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
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 = δqδx ˆ x ◮ Polarization is the density of induced dipoles: δxδa = δq δ� p � P = δa ˆ x
3 Bound charge due to polarization ◮ Charge density in infinitesimal chunk ρ b = δq 1 − δq 2 δaδx δq 1 δa − δq 2 δa = δx = P x ( x ) − P x ( x + δx ) δx = − ∂P x ∂x ◮ Similarly accounting for y and z components: ρ b = −∇ · � P
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 P = χ e ǫ 0 � � M = χ m � � E H D = (1 + χ e ) ǫ 0 � � B = (1 + χ m ) µ 0 � � E H ǫ = (1 + χ e ) ǫ 0 µ = (1 + χ m ) µ 0 ◮ Anisotropic dielectric: � χ e · ǫ 0 � P = ¯ E with susceptibility tensor ¯ χ e ◮ Nonlinear dielectric: � P = χ e ( E ) ǫ 0 � E
5 Capacitance q ◮ Gauss’s law: E = ǫ 0 A qd ◮ Potential difference V = d · E = ǫ 0 A ◮ Therefore stored charge per potential C ≡ q V = ǫ 0 A d ◮ Material produces polarization � P ◮ Corresponding bound charge density δq A = P on surface ◮ Gauss’s law: E = q − δq ǫ 0 A − P q ǫ 0 A = ǫ 0 ◮ q = A ( ǫ 0 E + P ) = AD = AǫE = Aǫ d V ◮ Therefore C = Aǫ d (increases by ǫ/ǫ 0 )
6 Sources of polarization ◮ Need to know constitutive relation � P = χ e ǫ 0 � E ◮ i.e. need χ e ≡ 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 χ e and α is χ e ǫ 0 = Nα , where N is number density of atoms / molecules? ◮ Almost, but not quite: local field differs from macroscopic field!
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 ρ b = −∇ · � P = 0 ◮ Except at edge of cavity, where it changes abruptly ◮ Bound surface charge density σ = − � P · ˆ n ◮ Corresponding electric field (at center): � 1 P cos θ - 2 πr 2 d cos θ δE loc = 4 πǫ 0 r 2 cos θ - + � �� � − 1 - + d A + -- = P + 3 ǫ 0 + � ◮ Net local field � E loc = � P E + 3 ǫ 0
8 Clausius-Mossoti relation p = α � ◮ Induced dipole moment � E loc ◮ Polarization density � p = Nα � P = N� E loc ◮ Relation between χ e and α : � � � P P = Nα � � � E loc = Nα E + 3 ǫ 0 � � 1 − Nα � = Nα � P E 3 ǫ 0 Nα � P > Nα ǫ 0 χ e ≡ = 1 − Nα ǫ 0 ǫ 0 3 ǫ 0 i.e. local field enhances response ◮ Dielectric constant (Clausius-Mossoti relation): 1 + 2 Nα ǫ − ǫ 0 = Nα 3 ǫ 0 ǫ = ǫ 0 (1 + χ e ) = ǫ 0 ⇔ 1 − Nα ǫ + 2 ǫ 0 3 ǫ 0 3 ǫ 0
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: x + qEe − iωt m ¨ x = − kx − γ ˙ ◮ Solve in frequency domain: qE x = k − iγω − mω 2 ◮ Polarizability (Lorentz oscillator model): q 2 α = p E = qx E = k − iγω − mω 2 ◮ 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
10 Frequency response of polarization 5 Re( α ) Im( α ) 4 3 2 α [ q 2 / k ] 1 0 -1 -2 -3 0 0.5 1 1.5 2 ω [( k/m ) 1/2 ] ◮ Strength of response ∝ q 2 /k � ◮ Frequency range set by resonant frequency ω 0 = k/m ◮ Width of resonance set by damping γ/m
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 · � P (cos θ ) ∝ exp − ( − � E ) = exp pE cos θ k B T k B T ◮ Thermal average dipole moment: � 1 � 1 − 1 d cos θp cos θ exp pE cos θ − 1 d cos θP (cos θ ) p cos θ k B T p = = � 1 � 1 − 1 d cos θ exp pE cos θ − 1 d cos θP (cos θ ) k B T � 1 − 1 d cos θ exp pE cos θ � 1 ∂ k B T = k B T∂ d cos θ exp pE cos θ ∂E k B T = log � 1 − 1 d cos θ exp pE cos θ ∂E k B T − 1 k B T
12 Dipole rotational response � 1 p = k B T ∂ d cos θ exp pE cos θ ∂E log k B T − 1 � k B T � �� = k B T ∂ exp pE k B T − exp − pE ∂E log pE k B T � � = k B T ∂ log sinh pE k B T − log E + const. ∂E � � k B T coth pE p k B T − 1 = k B T E � � 2 � � k B T = E p 2 k B T pE coth pE k B T − k B T pE pE pE p 2 kBT coth k B T − 1 α = � � 2 k B T pE k B T
13 Field response of polarization 0.5 0.4 1/3 α [ p 2 /( k B T )] 0.3 0.2 1/x 0.1 0 0 1 2 3 4 5 6 7 8 9 10 Ε [ k B T/p ] p 2 ◮ At low fields α = 3 k B T ◮ Response decays due to saturation at fields ∼ k B T p ◮ Maximum induced dipole = p (complete alignment)!
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
15 Insulation ◮ Metals: high conductivity ◮ Semiconductors: lower, but controllable conductivity ◮ Insulators: minimize conductivity σ [ (Ωm) − 1 ] Substance ρ [ Ωm ] 1 . 59 × 10 − 8 6 . 30 × 10 7 Silver 1 . 68 × 10 − 8 5 . 96 × 10 7 Copper 5 . 6 × 10 − 8 1 . 79 × 10 7 Tungsten 2 . 2 × 10 − 7 4 . 55 × 10 6 Lead 4 . 2 × 10 − 7 2 . 38 × 10 6 Titanium 6 . 9 × 10 − 7 1 . 45 × 10 6 Stainless steel 9 . 8 × 10 − 7 1 . 02 × 10 6 Mercury 5 − 8 × 10 − 4 1 − 2 × 10 3 Carbon (amorph) 4 . 6 × 10 − 1 Germanium 2.17 6 . 4 × 10 2 1 . 56 × 10 − 3 Silicon 1 . 0 × 10 12 1 . 0 × 10 − 12 Diamond 7 . 5 × 10 17 1 . 3 × 10 − 18 Quartz 10 23 − 10 25 10 − 25 − 10 − 23 Teflon
16 Insulators: band structure criteria ◮ What do we need to minimize conductivity? ◮ Large band gap ( � 100 k B T ) to reduce intrinsic conductivity N c N v exp − E g � σ i = e ( µ e + µ h ) n i = e ( µ e + µ h ) 2 k B T (mobilities and N c/v change only by one-two orders) ◮ Few ionizable dopants or defects to produce free carriers ( σ ∝ N a , N d ) ◮ First metric for insulator: small σ ◮ Second metric: dielectric breakdown strength E br ◮ Field beyond which material starts conducting (a nonlinear response)
17 Electronic breakdown: perfect crystals ◮ Perfect material: still few intrinsic carriers ◮ Apply electric field E , carrier can pick up energy e E λ before scattering ◮ If e E λ > E g , 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, E g ∼ 5 eV ⇒ E � 10 8 V/m. ◮ Additional sources of free carriers for breakdown: ◮ Injection from surfaces ◮ Photo-ionization by light / radiation / cosmic rays
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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