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

Chapter 2: Dielectric Properties Flashcards

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
Microscopic Theory of ε(ω) | (Overview: 4 points)
* Solids classified based upon response to EM wave * Dielectric * Paraelectric * Ferro-/Antiferroelectric
26
Dielectric Solids | (4 points)
* 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
27
Paraelectric Solids | (5 points)
* 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
28
Ferro-/Antiferroelectric Solids
Experience *spontaneous polarization*
29
Electronic Polarization | (Overview: 4 points)
* 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
30
Lorentz Oscillator Model | (Assumptions: 6 points)
* Ignore local electric field effects * Negative electron bound to positive atomic core is _perfect hamonic oscillator_ * Excited by ***E***(*t* ) = ***E***o*e*-iωt → _forced, damped hamonic oscillator_ * Equation of Motion (see below) * Stationary solution *x*(*t* ) = (-*e*/*m*) *f* (ω) ***E***(*t* ) * *f* (ω) is complex function
31
Lorentz Oscillator Model | (Atomic Polarizability: 2 points)
* Displacement *x* → dipole moment ***p***el = −*e**x*** = *ε*o*α**E*** * ​Equating coefficients
32
Lorentz Oscillator Model | (Dielectric Function: 3 points)
* Look at connection between macro/micro descriptions * ***P*** = *ε*o*n*v*α*(*ω*)***E*** = *ε*o*χ*(*ω*)***E*** * *ε*(*ω*) = 1 + *χ*(*ω*) = 1 + *n*v*α*(*ω*) = *ε*r + *iε*i
33
Lorentz Oscillator Model | (Dielectric Function: graph)
34
Ionic Polarization | (Overview: 3 points)
* 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 |***k***photon| ≈ 0 → only optical phonons with *q* = 0
35
Ionic Polarization | (Effect of ***E***loc [Overview]: 3 points)
* Additional force either strengthens or weakens restoring force * Like electronic polarization, system is *forced, damped harmonic oscillator* * Damping comes from heat dissipation
36
Ionic Polarization | (Effect of Eloc [Longitudinal Modes]: 3 points)
* ***E***loc is opposite relative ion displacement → enhanced restoring force * Longitudinal eigenfrequency increases ωL \> ωo
37
Ionic Polarization | (Effect of ***E***loc [Transverse Modes]: 2 points)
* ***E***loc is same direction as relative ion displacement → diminished restoring force * Transverse eigenfrequency decreases ωT \< ωo
38
Ionic Polarization | (Lyddane-Sachs-Teller Relation: 4 points)
* 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
Ionic Polarization | (Dielectric Function: graph)
40
Characteristic Eigenfrequencies | (2 points)
* Ions: IR region (1014 Hz) * Electrons: UV region (1016 Hz)
41
Ionoic Polarization | (Forced Oscillaton: 3 points)
* Lattice vibrations induced by external field * _Photon_: Transverse EM field → cannot excite longitudinal modes * _Electron_: High energy → can excite longitudinal modes
42
Optical Properties of Crystal | (Dielectric Function: 3 points)
* Photons can only interact with phonons near center of BZ (i.e. intersection with optical branch) * Consider only ***q*** = 0
43
Optical Properties of Crystals | (Dielectrifc Function Observations: 4 points)
* 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 \< ωUV → ε(ω) \> 0 → R \< 1
44
Optical Properties of Crystals | (Polaritons: 3 points)
* For ***q*** \> 0, apply results for ε(ω) to general dispersion ω2ε(ω) = c2q2 * Photon and phonon couple → _polariton_ * Difficult to excite
45
Optical Properties of Crystals | (Polariton Dispersion Relation: graph)
46
Orientation Polarization | (Static Polarization: 3 points)
* Paraelectric Materials have *permanent dipoles* * Potential energy given by (see below) * Only partial alignment *P*dip*E* \<\< *k*B*T*
47
Orientation Polarization | (Langevin-Debye Relation: 3 points)
* Relates ***P***dip to thermodynamic average of dipole (see below) * One can show: = pdipE/(3kB​)
  • Leads to Curie Law of MAgnetic Susceptability
48
Orientation Polarization | (Curie Law of Magnetic Susceptibility)
49
Orientation Polarization | (Frequency Dependence: 3 points)
* Dipoles cannot follow arbitrarily high AC-fields → *χ*dip(*ω*) decreases as *ω* increases * Introduce relaxation time *τ* to account for dipole inertia (see below) * Solutions: ***P***dip(*t* ) = ***P***dip(*ω*)[1 − exp(−*t* /*τ*)]
50
Orientation Polarization | (Debye Formula: 4 points)
* Assume *χ*dip(*ω*) in ***P***dip(*ω*) = *ε*o*χ*dip(*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
Orientation Polarization | (Debye Formula: graph)
52
Full Dielectric Response of an Insulator
53
Dielectrics of Free Electron Gas | (Assumptions: 5 points)
* Free electron gas in AC-field (Drude model) * Large wavelength *q* → 0 * EOM (see below) * Solution * *x*(*t* *)* = -e/m\* f(ω) ***E***
54
Dielectrics of Free Electron Gas | (Polarization: 3 points)
Relative displacement of conducting electrons and ions leads to polarization (and restoring force)
55
Dielectrics of Free Electron Gas | (Dielectric Function: 3 points)
* Total dielectric function sum of bound electrons *χ*el and conducting electrons *χ*c * ​Plasma frequency ωp2 ∝ nV
56
Dielectrics of Electron Gas | (Dielectric Function [Real/Imaginary])
57
Optical Properties of Metals | (Regimes: 3 points)
1. Low-Frequency Regime 2. Relaxation Regime 3. Transparent Regime
58
Optical Properties of Metals | (Low-Frequency Regime: 3 points)
* *ω* \<\< *ω*p → *ωτ \<\<* 1 → C \>\> 1 * *ε*r = -*ε*i*ωτ* → *ε*r \<\< *ε*i → lots of screening (i.e. losses much greater * Near-perfect reflection (metals make good mirrors)
59
Optical Properties of Metals | (Low-Frequency Range [Hagen-Rubens Relation]: 2 points)
As *σ*(0) increases, *R* → 1
60
Optical Properties of Metals | (Low-Frequency Regime [Skin Depth]: 2 points)
More conductive → smaller skin depth, i.e. more screening
61
Optical Properties of Metals | (Relaxation Regime: 4 points)
* 1/*τ* \< *ω* \< *ω*p → electrons can follow, but scattering becomes important * *ωτ* term dominates in denominator * *ε*(*ω*) ≈ *ε*r ≈ −*ε*el(*ω*)*ω*p2/*ω*2 * Looking at complex index of refraction → K ≈ 1/*ω*
62
Optical Properties of Metals | (Transparent Regime: 4 points)
* *ω* \>\> *ω*p → *ω*p/*ω \<\<* 1 * *ε*r(*ω*) ≈ *ε*el(*ω*) and *ε*i(*ω*) ≈ 0 → *ε*(*ω*) ≈ *ε*r(*ω*) * Complex index of refraction ≈ 1 * Metals become transparent above *ω*p (UV range)
63
Optical Properties of Metal | (Dielectric Function: graph)
64
Longitudinal Plasma Oscillations | (Overview: 6 points)
* _Recall_: Case for relflection → *ε* \< 0 * _Insulators_: *ω*L \< *ω* \< *ωT* * _Metals_: 0 \< *ω* \< *ωP* * Suggests * *ω*T = 0 * *ω*L = *ω*P
65
Longitudinal Plasma Oscillations | (Take-Away: 4 points)
* No transverse modes for electron gas * Longitudinal restoring force is ***F*** = -*e**E*** * ***​***Harmonic oscillator with plasma frequency *ω*P * Collective longitudinal oscillation of conducting electrons is a _plasmon_​
66
Longitudinal Plasma Oscillations | (Plasmon: 5 points)
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
Forced Transverse Plasma Oscillations | (Plasmon-Polariton [Overview]: 4 points)
* _Recall_: General expression for dispersion of photons * *ω*2 = *c*2*q*2/ε(*ω)* * ​Consider relaxation regime * Leads to _plasmon-polariton_
68
Forced Transverse Plasma Oscillation | (Plasmon-Polariton: 3 points)
* Transverse EM wave in metal * Coupling of photon and phonon
69
Plasma Oscillations | (Plasmon Dispersion: graph + 2 points)
* _Top_: Bulk plasmon * _Bottom_: Surface plasmon
70
Surface Plasmons | (Overview: 4 points + graph)
* 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
Surface Plasmon | (On Metal Surfaces: 5 points)
* *ε*air = 1 and *ε*metal \< 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
Surface Plasmon | (On Metal Surfaces [Kretschmann Configuration]: diagram)
73
Excitons | (Expectation v. Reality: 2 points)
* _Expectation_: Interband transitions in semiconductor at low *T* for *(h-bar)ω* \> *E*g * _Reality_: Excitation for *(h-bar)ω* \< *E*g → _excitons_
74
Excitons | (Properties: 4 points)
* Bound state between excited electron in conduction abnd and corresponding hole in valence band * Absorption energy (*h-bar*)*ω* = *E*g − *E*exciton * Can move through crystal as electron relaxes * Bosons
75
Exictons | (Absoroption Profile: graph)
76
Excitons | (Frenkel Excitions: 2 points)
* Typical for material with strongly bounded electrons * Smaller separation distance
77
Excitons | (Mott-Wannier Excitons: 2 points)
* Typical for matierals with weakly bonded electrons * Spatial distance of electron-hole pair \>\> lattice spacing
78
Electron-Electron Interactions | (Static Screening: 2 points + figure)
* Place positive test charge into system * Charges rearrange to create *P-* and *D-*fields, which cancel *ε****E*** = (***D*** - ***P*** ) = 0
79
Electron-Electron Interactions | (Thomas-Fermi Screening [Assumptions]: 4 points)
* Semi-classical theory * Slowly varying potential with respect to Fermi length * Constant chemical potential * Low temperature
80
Electron-Electron Interactions | (Thomas-Fermi Screening [Take-Away]: 2 points)
* *ε*(***q*** ) = 1 + *k*f2/*q*2 * For *q* → 0, *ε* → ∞ and full screening for non-zero *φ*ext
81
Ferroelectricity | (Overview: 4 points)
* Materials with spontaneous polarization *P*s that sets in below critical temperature, _Curie temperature_ *T*C, without extenal field * For *T* \> *T*C → called _paraelectric_ * Can have either (anti-)ferroelectricity by switching external field * Depends on structure
82
Ferroelectricity | (Pyroelectric)
Material where static field from spontaneous polarization is larger than switching field
83
Ferroelectricity | (Requirement for *P*s: 2 points + diagram)
* _Polar axis_: crystall cannot be roated 180 degrees into itself for any axis perpendicular to polar axis * Existence *does not* guarantee ferroelectricity
84
Ferroelectricity | (Piezoelectric Material)
Material where spontaneous polarization sets in when squeezed
85
Ferroelectricity | (Order Parameter: 2 points)
* Phase transition defined according to an order parameter * Here, spontaneous polarization *P*s
86
Ferroelectricity | (Landau Theory [Overview]: 4 points)
* 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
Ferroelectricity | (Landau Theory [Assumptions and Take-Away]: 2 points)
* Close to phase transition, *P*s very small → expansion of free-energy density *f*(***P***s, *T*, ***E*** ) around ***P***s (see below) * For ferroelectric, *a*2 = *γ*(*T* − *T*c) switches sign for finite *T*o ≤ *T*c
88
Ehrenfest Thermodynamic Classification Scheme for Phase Transitions | (First-Order)
First derivative of thermodynamic potential has discontinuity at phase transition
89
Ehrenfest Thermodynamic Classification Scheme for Phase Transitions ​(*nth*-Order: 4 points)
*nth*-derivative of thermodynamic potential has discontinuity at finite value * e.g. Second-order * *P*s = *∂F*/*∂E* changes continuously * *χ*ij = *∂*2*F*/*∂**E*2 has discontinuity
90
Classification of Ferroelectrics (5 points)
* _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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