SLIDE 1
MTLE-6120: Advanced Electronic Properties of Materials Insulating materials: dielectrics, ferroelectrics, piezoelectrics
Contents: ◮ Polarization contributions ◮ Frequency and field-dependent response ◮ Dielectric breakdown ◮ Symmetry breaking and spontaneous polarization Reading: ◮ Kasap: 7.1 - 7.12
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SLIDE 2 Materials in electric fields
◮ All materials composed of charges: electrons and nuclei ◮ Charges pulled along/opposite electric field with force q E ◮ Charges separated in each infinitesimal chunk of matter ⇒ dipoles ◮ Induced dipole moment: δ p = δqδxˆ x ◮ Polarization is the density of induced dipoles:
δ p δxδa = δq δa ˆ x
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SLIDE 3 Bound charge due to polarization
◮ Charge density in infinitesimal chunk ρb = δq1 − δq2 δaδx =
δq1 δa − δq2 δa
δx = Px(x) − Px(x + δx) δx = −∂Px ∂x ◮ Similarly accounting for y and z components: ρb = −∇ · P
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SLIDE 4 Constitutive relations
◮ Material determines how P (and hence D) depends on E ◮ Material determines how M (and hence H) depends on B ◮ Simplest case: linear isotropic dielectric
E
H
E
H ǫ = (1 + χe)ǫ0 µ = (1 + χm)µ0 ◮ Anisotropic dielectric: P = ¯ χe · ǫ0 E with susceptibility tensor ¯ χe ◮ Nonlinear dielectric: P = χe(E)ǫ0 E
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SLIDE 5 Capacitance
◮ Gauss’s law: E =
q ǫ0A
◮ Potential difference V =d · E =
qd ǫ0A
◮ Therefore stored charge per potential C ≡ q V = ǫ0A d ◮ Material produces polarization P ◮ Corresponding bound charge density
δq A = P on surface
◮ Gauss’s law: E = q−δq
ǫ0A = q ǫ0A − P ǫ0
◮ q = A(ǫ0E + P) =AD = AǫE = Aǫ
d V
◮ Therefore C = Aǫ
d (increases by ǫ/ǫ0)
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SLIDE 6
Sources of polarization
◮ Need to know constitutive relation P = χeǫ0 E ◮ i.e. need χe ≡ induced dipole / volume per unit field E ◮ Various sources:
◮ Electrons bound to atoms (ala HW1): electronic polarizability ◮ Dispalcement of ions in an ionic solid ◮ Rotations of dipoles in a dipolar material
◮ Define polarizability α ≡ dipole moment induced per unit field E in atom / molecule ◮ Relation between χe and α is χeǫ0 = Nα, where N is number density of atoms / molecules? ◮ Almost, but not quite: local field differs from macroscopic field!
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SLIDE 7 Lorenz equation
◮ Each atom / molecule feels effect of field E ◮ It additionally feels effect of field produced by surrounding atoms / molecules ◮ Surrounding molecules have P except in a ‘cavity’ containing the atom ◮ For simplicity assume spherical cavity ◮ Bound charge due to polarization ρb = −∇ · P = 0 ◮ Except at edge of cavity, where it changes abruptly ◮ Bound surface charge density σ = − P · ˆ n ◮ Corresponding electric field (at center): δEloc = 1
−1
2πr2d cos θ
P cos θ 4πǫ0r2 cos θ = P 3ǫ0 ◮ Net local field Eloc = E +
3ǫ0
+ + + + +
SLIDE 8 Clausius-Mossoti relation
◮ Induced dipole moment p = α Eloc ◮ Polarization density P =N p = Nα Eloc ◮ Relation between χe and α:
Eloc = Nα
3ǫ0
3ǫ0
E χe ≡
ǫ0 =
Nα ǫ0
1 − Nα
3ǫ0
> Nα ǫ0 i.e. local field enhances response ◮ Dielectric constant (Clausius-Mossoti relation): ǫ = ǫ0(1 + χe) = ǫ0 1 + 2Nα
3ǫ0
1 − Nα
3ǫ0
⇔ ǫ − ǫ0 ǫ + 2ǫ0 = Nα 3ǫ0
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SLIDE 9
Electronic / ionic polarization
◮ Charges: either electrons or ions in equilibrium position ◮ Displacement produces restoring forces: effective spring constant k ◮ Equation of motion in electric field: m¨ x = −kx − γ ˙ x + qEe−iωt ◮ Solve in frequency domain: x = qE k − iγω − mω2 ◮ Polarizability (Lorentz oscillator model): α = p E = qx E = q2 k − iγω − mω2 ◮ Qualitatively similar behavior for electrons and ions ◮ Electrons: high k and small m ⇒ smaller α over larger ω range ◮ Ions: small k and large m ⇒ greater α over smaller ω range
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SLIDE 10 Frequency response of polarization
1 2 3 4 5 0.5 1 1.5 2 α [q2/k] ω [(k/m)1/2] Re(α) Im(α)
◮ Strength of response ∝ q2/k ◮ Frequency range set by resonant frequency ω0 =
◮ Width of resonance set by damping γ/m
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SLIDE 11 Dipolar polarization
◮ What if molecules in solid have built-in dipole p? ◮ In response to field, they can produce additional induced dipole (electronic/ionic) ◮ Additionally, they can rotate to align with the field! ◮ Without field, dipoles in random direction ⇒ average = 0 ◮ With field, probability of dipole with given cos θ with respect to field P(cos θ) ∝ exp −(− p · E) kBT = exp pE cos θ kBT ◮ Thermal average dipole moment: p = 1
−1 d cos θP(cos θ)p cos θ
1
−1 d cos θP(cos θ)
= 1
−1 d cos θp cos θ exp pE cos θ kBT
1
−1 d cos θ exp pE cos θ kBT
= kBT
∂ ∂E
1
−1 d cos θ exp pE cos θ kBT
1
−1 d cos θ exp pE cos θ kBT
= kBT∂ ∂E log 1
−1
d cos θ exp pE cos θ kBT
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SLIDE 12 Dipole rotational response
p = kBT ∂ ∂E log 1
−1
d cos θ exp pE cos θ kBT = kBT ∂ ∂E log kBT pE
kBT − exp −pE kBT
∂E
kBT − log E + const.
kBT coth pE kBT − 1 E
kBT
pE coth pE kBT − kBT pE 2 α = p2 kBT
pE kBT coth pE kBT − 1
kBT
2
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SLIDE 13 Field response of polarization
0.1 0.2 0.3 0.4 0.5 1 2 3 4 5 6 7 8 9 10 α [p2/(kBT)] Ε [kBT/p] 1/3 1/x
◮ At low fields α =
p2 3kBT
◮ Response decays due to saturation at fields ∼ kBT
p
◮ Maximum induced dipole = p (complete alignment)!
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SLIDE 14
Sources of polarization summary
◮ Single material would have multiple contributions which add together ◮ Electronic: typically weakest, highest frequency (visible/ultraviolet) ◮ Ionic: stronger, till vibrational frequency range (infrared) ◮ Dipole: stronger still, till rotational frequency range (microwave/IR) ◮ Interfacial / defect charges: slowest response (if present)
Figure 7.18 from Kasap
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SLIDE 15
Insulation
◮ Metals: high conductivity ◮ Semiconductors: lower, but controllable conductivity ◮ Insulators: minimize conductivity Substance ρ [Ωm] σ [(Ωm)−1] Silver 1.59 × 10−8 6.30 × 107 Copper 1.68 × 10−8 5.96 × 107 Tungsten 5.6 × 10−8 1.79 × 107 Lead 2.2 × 10−7 4.55 × 106 Titanium 4.2 × 10−7 2.38 × 106 Stainless steel 6.9 × 10−7 1.45 × 106 Mercury 9.8 × 10−7 1.02 × 106 Carbon (amorph) 5 − 8 × 10−4 1 − 2 × 103 Germanium 4.6 × 10−1 2.17 Silicon 6.4 × 102 1.56 × 10−3 Diamond 1.0 × 1012 1.0 × 10−12 Quartz 7.5 × 1017 1.3 × 10−18 Teflon 1023 − 1025 10−25 − 10−23
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SLIDE 16 Insulators: band structure criteria
◮ What do we need to minimize conductivity? ◮ Large band gap ( 100kBT) to reduce intrinsic conductivity σi = e(µe + µh)ni = e(µe + µh)
2kBT (mobilities and Nc/v change only by one-two orders) ◮ Few ionizable dopants or defects to produce free carriers (σ ∝ Na, Nd) ◮ First metric for insulator: small σ ◮ Second metric: dielectric breakdown strength Ebr ◮ Field beyond which material starts conducting (a nonlinear response)
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SLIDE 17
Electronic breakdown: perfect crystals
◮ Perfect material: still few intrinsic carriers ◮ Apply electric field E, carrier can pick up energy eEλ before scattering ◮ If eEλ > Eg, carrier has enough energy to produce more electron-hole pairs ◮ New electron-hole pairs accelerated by field, produce further e-h pairs ⇒ cascade / avalanche ◮ Characteristic field scale: λ ∼ 50 nm, Eg ∼ 5 eV ⇒ E 108 V/m. ◮ Additional sources of free carriers for breakdown:
◮ Injection from surfaces ◮ Photo-ionization by light / radiation / cosmic rays 17
SLIDE 18
Breakdown mechanisms: materials with defects
◮ Thermal breakdown
◮ Field amplified near some defects in material ◮ Small conduction / high-frequency losses ⇒ local heating ◮ Increased temperature ⇒ higher conductivity ◮ More current, more heating ⇒ thermal runaway
◮ Electromechanical breakdown
◮ Electric field produces stresses ◮ Stresses enough to cause mechanical breakdown ◮ Current pathways open up through cracks / physical contact across thin films
◮ Discharges in porous materials
◮ Air gaps in material: lower dielectric strength ◮ Discharge in the gas: current pathway bypassing solid
◮ (See Kasap 7.6 for more details)
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SLIDE 19
Capacitor materials
◮ Get high capacitance by
◮ High area ◮ Small plate separation ◮ High dielectic constant
◮ Ceramic capacitors: film of high-dielectric ceramics (MHz)
◮ Small capacitances than types below ◮ High breakdown voltage (KV) and frequency response
◮ Electrochemical capacitors:
◮ High capacitance due to small separation (∼ 1 nm between electrode and electrolyte) ◮ Lower breakdown voltage (10-100 V) and frequency response (KHz)
◮ Supercapacitors:
◮ Electrochemical capacitor in pseudocapacitance regime ◮ Store charge by reversible redox reaction of surface species ◮ Even smaller breakdown voltage (< 10 V) and frequency response (10 Hz)
◮ (See Kasap 7.7 for more details)
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SLIDE 20 Piezo-electricity
◮ Centro-symmetric ( r → − r) crystals: no built-in dipole moment ◮ Non centro-symmetric crystals: can have built-in dipole per unit cell ◮ Strain symmetric material: no net dipole ◮ Broken symmetry: induced dipole ◮ Piezoelectric coefficients for induced dipole Pi = dij Tj
◮ Converse: applied field produces strain Sj = dijEi ◮ Same coefficients in both equations: why? 1 2Ei · Pi
= 1 2Tj · Sj
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SLIDE 21 Piezoelectric applications
◮ Spark generators: induce high field using stress ◮ Crystal oscillators
◮ Electric field coupled to mechanical
◮ Design quartz crystal dimensions to set resonant frequency ◮ Pickup oscillations as voltage with same frequency ◮ Use electronic circuit to amplify ◮ Stable sharp frequencies compared to electrical (LC) oscillators
◮ Precise motion at the atomic scale: use piezo-strain for STM / AFM tip motion
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SLIDE 22
Ferroelectricity and pyroelectricity
◮ Non-centrosymmetric crystals: built in P per unit cell ◮ Crystal grains could have P in random directions: dielectric ◮ Ferroelectric phase transition with critical temperature Tc ◮ Below Tc, crystal grains (domains) align: finite P without applied E ◮ Non-centrosymmetric ⇒ must also be piezeoelectric ◮ Analgous to ferromagnetic materials (discussed in more detail next) ◮ Potential applications as electronic memory devices ◮ Dipole of ferroelectric material changes with T ◮ Pyroelectric coefficient dP/dT
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