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Advanced Experimental Physics II > Chapter 2: Dielectric Properties > Flashcards

Chapter 2: Dielectric Properties Flashcards

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1
Q

Goal of Studying Dielectrics Properties

(3 points)

A

Determine relation between microscoptic and macroscopic descriptions of how solids as a whole respond to electric field

  • Microscopic VIew: photon creates phonon and/or electron-hole
  • Macroscopic View: absorption described by Maxwell’s Equations and optical absoroption coefficient
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2
Q

Macroscopic Electrodynamics

(Assumptions: 2 points)

A
  • Linear response
    • i.e. small fields, no saturation effects, etc
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3
Q

Response of solid to electric field

(3 points)

A
  • Depends on charge mobility
    • Metals: free charge carriers → screening of E at short distances
    • Insulators: small electron motion (polarization) → screening of E at long distances
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4
Q

Insulator response to
E = E*exp[i(qr-ωt)]

A

Small displacement induces polarization

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5
Q

Dielectric Displacement Field

(2 points)

A
  • Accounts for effects of free and bound charges in material (only free charges are source)
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6
Q

Dielectric Function

(Insulators: 3 points)

A
  • Usually, wavelength of EM wave λ >> a lattice constant
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7
Q

Metal response to
E = E*exp[i(qr-ωt)]

A

Electric current forms

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8
Q

Maxwell’s Equations

A
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9
Q

Generalized conductivity

(5 points)

A
  • Consider ∇ × H with J = σE and D(ω) = εoε(ω)E
    • ​Real part: due to free charges; dominates in metals
    • Imaginary part: due to polarized charges
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10
Q

Generalized dielectric constant

(3 points)

A
  • Real part: due to polarization
  • Imaginary part: due to free charges
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11
Q

Kramer-Kronig Relationship

(Assumptions: 2 parts)

A
  • Linear response
    • E(ω) ∝ P(ω) → χ(ω) and ε(ω) are linear functions
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12
Q

Kramer-Kronig Relationship

(Take-Away)

A
  • Measuring either real or imaginary part over entire sepctrum allows recovering the other part
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13
Q

Absorption, Transmission, and Reflection of EM Radiation

(Overview)

A

There exists connection between dielectric properties and optical parameter of solid

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14
Q

Absorption, Transmission, and Reflection of EM Radiation

(Assumptions: 3 points)

A
  • Uncharged solid ∇ · D = ρfree = 0
  • Non-magnetic material µ = 0
  • From MW-equations (see below)
    • Should by vph2 in denominator
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15
Q

Absoroption, Transmission, and Reflection of EM Radiation

(Propagation (phase) Velocity)

A

Speed of EM wave in material

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16
Q

Absorption, Transmission, and Reflection of EM Radiation

(Complex Refractive Index: 2 points)

A
  • n(ω): refractice index
  • K(ω): absorption coefficient
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17
Q

Absorption, Transmission, and Reflection of EM Radiation

(Complex Wave Vector)

A
  • Recall: Dispersion relation ω = ck/n
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18
Q

Absorption, Transmission, and Reflection of EM Radiation

(Complex refractive index: graph)

A
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19
Q

Absorption, Transmission, and Reflection of EM Radiation

(Take-Away)

A
  • Solution with complex wave vector is damped wave (see below)
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20
Q

Absorption, Transmission, and Reflection of EM Radiation

(Take-Away: graph)

A
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21
Q

Absorption, Transmission, and Reflection of EM Radiation

(Take-Away: Absorption Coefficient)

A

Provides relationship between experimentall measureable quantity K and dielectric property of material ε

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22
Q

The Local Electric Field Eloc

A

Due to shielding, local electric field generally not identical to external field

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23
Q

The Local Electric Field

(The Dipole Electric Field Edip: 5 points)

A
  • Lorentz field EL
  • Depolarization field EN
  • Depolarization factor N
  • Macroscopic polarization P
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24
Q

The Local Electric Field

(Diagram)

A
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25
Q

Microscopic Theory of ε(ω)

(Overview: 4 points)

A
  • Solids classified based upon response to EM wave
    • Dielectric
    • Paraelectric
    • Ferro-/Antiferroelectric
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26
Q

Dielectric Solids

(4 points)

A
  • Can experience electronic or ionic polarization
    • Typically dominate in insulators
    • Both cases, restoring forces leads to forced harmonic oscillator with characterisitc eigenfrequency​
  • ​Metals are special case because no restoring force
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27
Q

Paraelectric Solids

(5 points)

A
  • Solids with permanent electric dipoles
    • Can experience orientation polarization
      • Increases as temperature (external field) decreases (increases)
  • Solid must have asymmetric molecules
  • Characteristic frequencies small due to large mass
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28
Q

Ferro-/Antiferroelectric Solids

A

Experience spontaneous polarization

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29
Q

Electronic Polarization

(Overview: 4 points)

A
  • Interaction of EM wave with solid leads to excitation of electron, i.e. it becomes polarized
    • ​Intraband transitions
    • Interband transitions
  • Consider insulators, with only interband treansitions
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30
Q

Lorentz Oscillator Model

(Assumptions: 6 points)

A
  • Ignore local electric field effects
  • Negative electron bound to positive atomic core is perfect hamonic oscillator
  • Excited by E(t ) = Eoe-<em>iωt</em> → forced, damped hamonic oscillator
  • Equation of Motion (see below)
  • Stationary solution x(t ) = (-e/m) f (ω) E(t )
    • f (ω) is complex function
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31
Q

Lorentz Oscillator Model

(Atomic Polarizability: 2 points)

A
  • Displacement x → dipole moment pel = −ex = εoαE
    • ​Equating coefficients
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32
Q

Lorentz Oscillator Model

(Dielectric Function: 3 points)

A
  • Look at connection between macro/micro descriptions
    • P = εonvα(ω)E = εoχ(ω)E
    • ε(ω) = 1 + χ(ω) = 1 + nvα(ω) = εr + iεi
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33
Q

Lorentz Oscillator Model

(Dielectric Function: graph)

A
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34
Q

Ionic Polarization

(Overview: 3 points)

A
  • Caused by oppositely charged ions oscillating against eachother (i.e. optical phonons)
  • Interaction of photon with ions of crystal ≡ collision between photon and optical phonon
    • From conservation of moments |kphoton| ≈ 0 → only optical phonons with q = 0
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35
Q

Ionic Polarization

(Effect of Eloc [Overview]: 3 points)

A
  • Additional force either strengthens or weakens restoring force
  • Like electronic polarization, system is forced, damped harmonic oscillator
    • Damping comes from heat dissipation
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36
Q

Ionic Polarization

(Effect of Eloc [Longitudinal Modes]: 3 points)

A
  • Eloc is opposite relative ion displacement → enhanced restoring force
  • Longitudinal eigenfrequency increases ωL > ωo
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37
Q

Ionic Polarization

(Effect of Eloc [Transverse Modes]: 2 points)

A
  • Eloc is same direction as relative ion displacement → diminished restoring force
  • Transverse eigenfrequency decreases ωT < ωo
38
Q

Ionic Polarization

(Lyddane-Sachs-Teller Relation: 4 points)

A
  • Only valid for q = 0
  • ε(0) ≡ static value (ω ≈ 0) → photon in IR-range (ω << ωion,o )
  • εstat ≡ value in visible range (ωion,o << ω << ωel,o )
39
Q

Ionic Polarization

(Dielectric Function: graph)

A
40
Q

Characteristic Eigenfrequencies

(2 points)

A
  • Ions: IR region (1014 Hz)
  • Electrons: UV region (1016 Hz)
41
Q

Ionoic Polarization

(Forced Oscillaton: 3 points)

A
  • Lattice vibrations induced by external field
    • Photon: Transverse EM field → cannot excite longitudinal modes
    • Electron: High energy → can excite longitudinal modes
42
Q

Optical Properties of Crystal

(Dielectric Function: 3 points)

A
  • Photons can only interact with phonons near center of BZ (i.e. intersection with optical branch)
    • Consider only q = 0
43
Q

Optical Properties of Crystals

(Dielectrifc Function Observations: 4 points)

A
  • Singularity ω = ωT → total absorption by crystal
  • Becomes zero at ω = ωL → agrees with longitudinal modes only possible for ε(ω) = 0
  • For ωT < ω < ωL → ε(ω) < 0 → total relfection because index of refraction n = ε1/2 is imaginary
  • For ωL < ω < ω<span>UV </span>→ ε(ω) > 0 → R < 1
44
Q

Optical Properties of Crystals

(Polaritons: 3 points)

A
  • For q > 0, apply results for ε(ω) to general dispersion ω2ε(ω) = c2q2
    • Photon and phonon couple → polariton
  • Difficult to excite
45
Q

Optical Properties of Crystals

(Polariton Dispersion Relation: graph)

A
46
Q

Orientation Polarization

(Static Polarization: 3 points)

A
  • Paraelectric Materials have permanent dipoles
    • Potential energy given by (see below)
      • Only partial alignment PdipE << kBT
47
Q

Orientation Polarization

(Langevin-Debye Relation: 3 points)

A
  • Relates Pdip to thermodynamic average of dipole (see below)
  • One can show: <cos> = <em>p</em>dip<em>E</em>/(3<em>k</em>B<em>​</em>)
    </cos><ul>
    <li>Leads to Curie Law of MAgnetic Susceptability </li>
    </ul></cos>
48
Q

Orientation Polarization

(Curie Law of Magnetic Susceptibility)

A
49
Q

Orientation Polarization

(Frequency Dependence: 3 points)

A
  • Dipoles cannot follow arbitrarily high AC-fields → χdip(ω) decreases as ω increases
    • Introduce relaxation time τ to account for dipole inertia (see below)
    • Solutions: Pdip(t ) = Pdip(ω)[1 − exp(−t /τ)]
50
Q

Orientation Polarization

(Debye Formula: 4 points)

A
  • Assume χdip(ω) in Pdip(ω) = εoχdip(ω)E(ω) has real and imaginary part
  • Solution has real and imaginary party (see below)
    • Real part: dipole alignment
    • Imaginary part: dielectric losses due to relaxation
51
Q

Orientation Polarization

(Debye Formula: graph)

A
52
Q

Full Dielectric Response of an Insulator

A
53
Q

Dielectrics of Free Electron Gas

(Assumptions: 5 points)

A
  • Free electron gas in AC-field (Drude model)
  • Large wavelength q → 0
  • EOM (see below)
  • Solution
    • x(t ) = -e/m* f(ω) E
54
Q

Dielectrics of Free Electron Gas

(Polarization: 3 points)

A

Relative displacement of conducting electrons and ions leads to polarization (and restoring force)

55
Q

Dielectrics of Free Electron Gas

(Dielectric Function: 3 points)

A
  • Total dielectric function sum of bound electrons χel and conducting electrons χc
    • ​Plasma frequency ωp2 ∝ nV
56
Q

Dielectrics of Electron Gas

(Dielectric Function [Real/Imaginary])

A
57
Q

Optical Properties of Metals

(Regimes: 3 points)

A
  1. Low-Frequency Regime
  2. Relaxation Regime
  3. Transparent Regime
58
Q

Optical Properties of Metals

(Low-Frequency Regime: 3 points)

A
  • ω << ωp → ωτ << 1 → C >> 1
  • εr = -εiωτ → εr << εi → lots of screening (i.e. losses much greater
    • Near-perfect reflection (metals make good mirrors)
59
Q

Optical Properties of Metals

(Low-Frequency Range [Hagen-Rubens Relation]: 2 points)

A

As σ(0) increases, R → 1

60
Q

Optical Properties of Metals

(Low-Frequency Regime [Skin Depth]: 2 points)

A

More conductive → smaller skin depth, i.e. more screening

61
Q

Optical Properties of Metals

(Relaxation Regime: 4 points)

A
  • 1/τ < ω < ωp → electrons can follow, but scattering becomes important
  • ωτ term dominates in denominator
  • ε(ω) ≈ εr ≈ −εel(ω)ωp2/ω2
  • Looking at complex index of refraction → K ≈ 1/ω
62
Q

Optical Properties of Metals

(Transparent Regime: 4 points)

A
  • ω >> ωp → ωp/ω << 1
  • εr(ω) ≈ εel(ω) and εi(ω) ≈ 0 → ε(ω) ≈ εr(ω)
  • Complex index of refraction ≈ 1
    • Metals become transparent above ωp (UV range)
63
Q

Optical Properties of Metal

(Dielectric Function: graph)

A
64
Q

Longitudinal Plasma Oscillations

(Overview: 6 points)

A
  • Recall: Case for relflection → ε < 0
    • Insulators: ω<span>L </span>< ω < ω<span>T</span>
    • Metals: 0 < ω < ω<span>P</span>
  • Suggests
    • ωT = 0
    • ωL = ωP
65
Q

Longitudinal Plasma Oscillations

(Take-Away: 4 points)

A
  • No transverse modes for electron gas
  • Longitudinal restoring force is F = -eE
    • ​Harmonic oscillator with plasma frequency ωP
    • Collective longitudinal oscillation of conducting electrons is a plasmon​
66
Q

Longitudinal Plasma Oscillations

(Plasmon: 5 points)

A

Collective longitudinal excitation of conducting electrons

  • Cannot be thermally or optically excited
  • Excite using fast, charged particles
    • Shoot through thin foil
    • Energy loss of transmitted electorn is excitation energy of plasmon
67
Q

Forced Transverse Plasma Oscillations

(Plasmon-Polariton [Overview]: 4 points)

A
  • Recall: General expression for dispersion of photons
    • ω2 = c2q2/ε(ω)
  • ​Consider relaxation regime
  • Leads to plasmon-polariton
68
Q

Forced Transverse Plasma Oscillation

(Plasmon-Polariton: 3 points)

A
  • Transverse EM wave in metal
  • Coupling of photon and phonon
69
Q

Plasma Oscillations

(Plasmon Dispersion: graph + 2 points)

A
  • Top: Bulk plasmon
  • Bottom: Surface plasmon
70
Q

Surface Plasmons

(Overview: 4 points + graph)

A
  • Can be excited at interfaces between material with opposite signs of dielectric function
  • Must use p-polarized light
  • Only one wave in each half-space with exponentially decaying amplitude perpendicular to surface
    • High concentration of electric field near surface
71
Q

Surface Plasmon

(On Metal Surfaces: 5 points)

A
  • εair = 1 and εmetal < 0 → surface plasmons
  • Opticall excite with Kretschmann Configuration
    • Detect dip in reflectivity when surface plasmon excited
    • Very sensitive to change in delectric constant
      • e.g. detect desorption process
72
Q

Surface Plasmon

(On Metal Surfaces [Kretschmann Configuration]: diagram)

A
73
Q

Excitons

(Expectation v. Reality: 2 points)

A
  • Expectation: Interband transitions in semiconductor at low T for (h-bar)ω > Eg
  • Reality: Excitation for (h-bar)ω < Eg → excitons
74
Q

Excitons

(Properties: 4 points)

A
  • Bound state between excited electron in conduction abnd and corresponding hole in valence band
  • Absorption energy (h-bar)ω = Eg − Eexciton
  • Can move through crystal as electron relaxes
  • Bosons
75
Q

Exictons

(Absoroption Profile: graph)

A
76
Q

Excitons

(Frenkel Excitions: 2 points)

A
  • Typical for material with strongly bounded electrons
  • Smaller separation distance
77
Q

Excitons

(Mott-Wannier Excitons: 2 points)

A
  • Typical for matierals with weakly bonded electrons
  • Spatial distance of electron-hole pair >> lattice spacing
78
Q

Electron-Electron Interactions

(Static Screening: 2 points + figure)

A
  • Place positive test charge into system
    • Charges rearrange to create P- and D-fields, which cancel εE = (D - P ) = 0
79
Q

Electron-Electron Interactions

(Thomas-Fermi Screening [Assumptions]: 4 points)

A
  • Semi-classical theory
  • Slowly varying potential with respect to Fermi length
  • Constant chemical potential
  • Low temperature
80
Q

Electron-Electron Interactions

(Thomas-Fermi Screening [Take-Away]: 2 points)

A
  • ε(q ) = 1 + kf2/q2
    • For q → 0, ε → ∞ and full screening for non-zero φext
81
Q

Ferroelectricity

(Overview: 4 points)

A
  • Materials with spontaneous polarization Ps that sets in below critical temperature, Curie temperature TC, without extenal field
  • For T > TC → called paraelectric
  • Can have either (anti-)ferroelectricity by switching external field
    • Depends on structure
82
Q

Ferroelectricity

(Pyroelectric)

A

Material where static field from spontaneous polarization is larger than switching field

83
Q

Ferroelectricity

(Requirement for Ps: 2 points + diagram)

A
  • Polar axis: crystall cannot be roated 180 degrees into itself for any axis perpendicular to polar axis
    • Existence does not guarantee ferroelectricity
84
Q

Ferroelectricity

(Piezoelectric Material)

A

Material where spontaneous polarization sets in when squeezed

85
Q

Ferroelectricity

(Order Parameter: 2 points)

A
  • Phase transition defined according to an order parameter
    • Here, spontaneous polarization Ps
86
Q

Ferroelectricity

(Landau Theory [Overview]: 4 points)

A
  • Examines free-energy density as function of spontaneous polarization to define phase transition
    • Can relate spontaneous polarizability and susceptability to free energy F = U - TS
87
Q

Ferroelectricity

(Landau Theory [Assumptions and Take-Away]: 2 points)

A
  • Close to phase transition, Ps very small → expansion of free-energy density f(Ps, T, E ) around Ps (see below)
    • For ferroelectric, a2 = γ(T − Tc) switches sign for finite To ≤ Tc
88
Q

Ehrenfest Thermodynamic Classification Scheme for Phase Transitions

(First-Order)

A

First derivative of thermodynamic potential has discontinuity at phase transition

89
Q

Ehrenfest Thermodynamic Classification Scheme for Phase Transitions
​(nth-Order: 4 points)

A

nth-derivative of thermodynamic potential has discontinuity at finite value

  • e.g. Second-order
    • Ps = ∂F/∂E changes continuously
    • χij = ∂2F/∂**E2 has discontinuity
90
Q

Classification of Ferroelectrics
(5 points)

A
  • Order-Disorder System
  • Displacive System
    • e.g. Figure below
      • If forces from local field > restoring forces → displacement occurs, i.e. freezing phonon
      • Polarization Catastrophe: Polarization is classically infinite, but the displacement stops due to anharmonic effects

Advanced Experimental Physics II (3 decks)

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