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Advanced Experimental Physics II > Chapter 4: Superconductivity > Flashcards

Chapter 4: Superconductivity Flashcards

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1
Q

Superconductivity

(Overview: 3 points)

A
  • Below critical temperature TC:
    • Perfect conductor: resistivity vanishes
    • Perfect diamagnet: repels magnet field inside with χ = −1
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2
Q

Superconductivity

(Types: 6 points)

A
  • 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
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3
Q

Basic Properties

(Perfect Conductor: 2 points)

A
  • resistivity vanishes below TC
  • Drops within ∆T = T(R = 0.9Rn)−T(R = 0.1Rn) for normal resistivity Rn
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4
Q

Basic Properties

(Perfect Conductor [Measurement]: 3 points + diagram)

A
  • Measure current inductively by measuring induced magnetic field
    • B ∝ I
    • Lifetimes upto 1014 s have beeb measured
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5
Q

Basic Properties

(Perfect Diamagnet [Meissner Effect: Overview]: 2 point + diagram)

A

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
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6
Q

Basic Properties

(Perfect Diamagnet [Meissner Effect: Supercooled with B = 0]: 4 points + diagram)

A
  • 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
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7
Q

Basic Properties

(Perfect Diamagnet [Meissner Effect: Supercooled with B > 0]: 6 points)

A
  • 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
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8
Q

Basic Properties

(Critical Field [Overview]: 2 point + diagram)

A
  • 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
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9
Q

Basic Properties

(Critical Field [Temperature Dependence])

A
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10
Q

Basic Properties

(Critical Field [Magnetization Inside]: 2 points + graph)

A
  • Bcth < 0 → can increase with extenral field
  • Bcth > 0 → magnetization breaks down
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11
Q

Basic Properties

(Critical Field [Field Inside]: 2 points + graph)

A
  • Bcth < 0 → external field shielded
  • Bcth > 0 → external field penetrates
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12
Q

Basic Properties

(Flux Quantization [Overview]: 2 points)

A
  • Experiments show that magnetic flux through a superconducting ring is an interger of a flux quantum (see below)
  • Experimental evidence of Cooper pairs
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13
Q

Basic Properties

(Flux Quantization [Experiment]: 1 point + graph)

A
  • Trap magnetic flux in supercooled lead tube and measure torque it causes on mirror
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14
Q

Thermodynamic Properties

(Overview: 3 points)

A
  • For type-I and -II superconductors in magnetic field, look at
    • Enthalpy
    • Entropy
    • Specific Heat
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15
Q

Thermodynamic Properties

(Type-I [Enthalpy]: 3 points + graph)

A
  • Phase transition at B = Bcth
    • For B < Bcth → change in enthalpy ∆G ∝ B2 (Meissner parabola)
    • B >= Bcth → enthalpy is that or normal state (≈ contst.)
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16
Q

Thermodynamic Properties

(Type-I [Enthalpy: Condensation Energy]: 3 points)

A
  • 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
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17
Q

Thermodynamic Properties

(Type-I [Entropy: Overview]: 2 points + graph)

A
  • Recall:
    • entropy of normal state Sn ∝ T
    • enthalpy of normal state gn ∝ T2
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18
Q

Thermodynamic Properties

(Type-I [Entropy: Take-Away]: 2points)

A
  • In B-field: first-order transition
  • No B-field: second-order transition
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19
Q

Thermodynamic Properties

(Type-I [Specific Heat: Overview]: 3 points)

A
  • Specific heat is measurable
  • Rutger’s formula gives different between normal and superconducting state ∆C = CN - CS
    • For T = TC → ∆C < 0
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20
Q

Thermodynamic Properties

(Type-I [Specific Heat: Take-Away]: 1 point + graph)

A
  • Heat capacity is greater in superconducting state
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21
Q

Thermodynamic Properties

(Type-I [Summary]: 3 points)

A
  • Enthalpy → condensation energy
  • Entropy → first-(second-)order phase transition with(out) B-field
  • Specific heat → superconducting state has higher heat capacity
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22
Q

Thermodynamic Properties

(Type-II [Overview]: 3 points)

A
  • 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
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23
Q

Thermodynamic Properties

(Type-II [Enthalpy]: 2 points + graph)

A
  • Bc1 < B < Bc2 → Shupnikov effect
    • enthalpy density increases slower than B2
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24
Q

Thermodynamic Properties

(Type-II [Entropy])

A
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25
Q

Thermodynamic Properties

(Type-II [Specific Heat])

A
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26
Q

London Equations

(Overview: 3 points)

A
  • Describe perfect conductor and perfect diamagneti with classical electrodynamics
  • Can account for Meissner effect
  • Can use to derive supercurrent carrier density ns
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27
Q

London Equations

(Assumptions: 4 points)

A
  • 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
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28
Q

London Equations

(First Equation: 2 points)

A

Relates supercurrent density Js = nsqsvs to external field E

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29
Q

London Equations

(Second Equation: 3 points)

A
  • 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
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30
Q

London Equations

(London Penetration Depth: 3 points)

A
  • from field screening equation ∇ × B = µoJ
    • _​London penetration depth_ λL2 = Λ/µo
31
Q

London Equations

(London Penetration Depth [Temperature Dependence])

A
  • Empirical relation
32
Q

London Equations

(Take-Away [Overview]: 2 points)

A
  • Consider superconducting material for x >= 0
  • There exist B-field penetration and supercurrent in superconducting material
33
Q

London Equations

(Take-Away [B-field]: 1 point + 1 graph)

A

Permits exponentially decaying B-fields (e.g. type-II)

34
Q

London Equations

(Take-Away [Supercurrent]: 1 point + 1 graph)

A

Supercurrent along surface

35
Q

London Equations

(Take-Away [Notes]: 4 points)

A
  • 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)
36
Q

Ginzburg-Landau Theory

(Overview: 5 points)

A
  • 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
37
Q

Ginzburg-Landau Theory

(Spatially Homogeneous, B = 0 [Assumptions]: 4 points)

A
  • |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)
38
Q

Ginzburg-Landau Theory

(Spatially Homogeneous, B = 0 [Take-Away]: 5 points)

A
  • Can relate the following to α, β
    • ​supercurrent carrier density ns
    • condensation energy ∆g
  • ​Meaning of α
    • condesation energy for forming one Cooper pair
39
Q

Ginzburg-Landau Theory

(Spatially Homogeneous, B = 0 [∆g vs |ψo(r)|]: 3 points + graph)

A
  • Left: above TC
  • MIddle: below TC
  • Right: complex |ψo(r)|
40
Q

Ginzburg-Landau Theory

(Spatially Varying, B > 0 [Assumptions])

A
  • Again, expand gs around |ψo(r)|2, but include terms for field repulsion and spartial variance of order
41
Q

Ginzburg-Landau Theory

(Spatially Varying, B > 0 [Take-Away]: 2 points)

A
  • 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
42
Q

Ginzburg-Landau Theory

(Penetration Depth: 2points)

A
  • λGL = λL/2
43
Q

Ginzburg-Landau Theory

(Coherence Length: 2 points)

A

Local perturbations in density decay expoentially with this characterstic length

44
Q

Ginzburg-Landau Theory

(Ginzburg-Landau Parameter: 3 points)

A
  • κ < 1 → type-I
  • κ > 1 → type-II
45
Q

Ginzburg-Landau Theory

(Spatial Ordering at Interface: 2 points + graph)

A
  • No external field
  • No diffusion of carriers
46
Q

Microscopic Description

(Goal)

A

Microscopic description of superconducting electron

47
Q

Microscopic Theory

(BCS Theory [Overview]: 5 points + equation)

A
  • 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
48
Q

Microscopic Theory

(BCS Theory [Take-Away]: 5 points + diagram)

A
  • 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
49
Q

Micoscopic Theory

(BCS Theory [Interaction Diagram])

A
50
Q

BCS Theory

(Cooper Pairs [Assumptions]: 3 points)

A
  • Free electron gas
  • T = 0
  • all states up to EF are filled
51
Q

BCS Theory

(Cooper Pairs [Overview]: 5 points + diagram)

A
  • 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)
52
Q

BCS Theory

(Cooper Pairs [Take-Away]: 2 points + equation)

A
  • Difference in energy of paired electrons ∆E makes Fermi sea unstable
  • Drives system to new phase → BCS groundstate
53
Q

BCS Theory

(BCS Groundstate: 3 points + graph)

A
  • 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
54
Q

BCS Theory

(BCS Groundstate [Condensation Energy])

A
55
Q

BCS Theory

(BCS Groundstate [Density of States]: 3 points + graph)

A
  • Divergent for Ek = ∆
  • Ek >> ∆ → DOS of free electron gas
  • Cooper pairs are δ-function at Ek = 0
56
Q

BCS Theory

(BCS State for T > 0: 4 points + graph)

A
  • As T → TC then ∆ → 0
    • Thermal breaking of Cooper pairs and exitation of quasi-particles
      • Hinders exchange of virtual phonons
  • Can show ∆(0) = 1.764kBTC
57
Q

Measuring ∆

(Indirectly: 4 points + 2 graphs)

A
  • 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)
58
Q

Measuring ∆

(Directly: 2 points)

A
  • Tunneling spectroscopy
  • Absorption spectroscopy
59
Q

Measuring ∆

(Optical Properties [Reflectivity]: 2 points + graph)

A
  • Reflectivity jumps from metal curve to R = 1 for ω ∝ 2∆
    • No absorption for ω < 2∆ → total reflectivity
60
Q

Measuring ∆

(Optical Properties [Spectral Weight]: 2 points + 2 graphs)

A
  • Spectral weight of σ1 jumps to δ-peak at ω = 0 after opening of gap
  • Induvtive response in σ2 → Meissner effect
61
Q

Measuring ∆

(Tunneling Spectroscopy [Setup]: 1 point + 1 diagram)

A
  • Two thin conducting regions separated by small insulating region
62
Q

Measuring ∆

(Tunneling Microscopy [Main Idea]: 3 points + graph)

A
  • 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
63
Q

Josephson Junction

(Overview: 4 points + diagram)

A
  • 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
64
Q

Josephson Junction

(Case U = 0: 3 points + graph)

A
  • 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
65
Q

Josephson Junction

(Case |U | > 0: 2 points + graph)

A
  • Quasi-particles begin tunneling
  • AC-Josephson effect: Current jumps from Josephson current IJ to finite value
66
Q

Josephson Junction

(B > 0: 2 points + diagram)

A
  • Use pair of Josephson junctions → SQUID
    • Allows precise magnetic flux measurements
67
Q

Unconventional Superconductors

(Overview: 6 points)

A
  • 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]
68
Q

Unconventional Superconductors

(Completing Order Classification: 3 points)

A
  • 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
69
Q

Cuprates

(Structure: 2 points + 2 diagrams)

A
  • Think of as stacks of Josephson junctions
  • Apex oxygen strongly controls TC
70
Q

Cuprates

(Electronic Structure: 5 points + 2 graphs)

A
  • 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
71
Q

Cuprates

(Electronic Structure [Doping]: 3 points)

A
  • Doping by replacing L3+ → C4+
    • ​Adds extra electrion to CuO layer, even though that’s not where dopant goes
      • Intrinsic modulation doped system
72
Q

Cuprates

(Fermi Surface: graph)

A
73
Q

Cuprates

(Spin Structure and Phase Diagram: 3 points + diagram)

A
  • t << U → no hopping
  • Uncertainty principle → virtual hopping
  • Spin fluctuations are possible “glue”
74
Q

Cuprates

(Superconducting Properties)

A

Highly anisotropic due to layers

Advanced Experimental Physics II (3 decks)

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