ssp_24 - 8 Dielectric properties D = ε0 P(8.1 D dielectric...

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Unformatted text preview: 8 Dielectric properties D = ε0( + P (8.1) D: dielectric displacement P= P: polarization 1 V ∑p i i (8.2) pi: electrical dipole moments ε0 = 8.854 × 10-12 As/Vm P = χε0( , ⇒ vacuum permittivity χ: dielectric susceptibility D = ε0(1+χ)( = ε0ε( (8.3) (8.4) ε: dielectric function, ε(ω) ⇒ optical properties in general 2nd rank tensor, scalar for isotropic media and cubic crystals. Insulators: P due to localized dipoles. Metals and semiconductors: also displacement of mobile electrons. Dielectric solids 8-1 a) Displacement of electrons with respect to atomic cores Always present but can be masked by other polarizations. Fig. 8.1 Illustration of polarization due to displacement of electrons in atoms with respect to atomic cores. b) Additional contribution in ionic crystal: relative displacement of cations and anions. Paraelectric solids Contain permanent electric dipoles (asymmetric polar molecules, e.g., H2O) ( = 0: dipoles oriented statistically ⇒ no net polarization ( ≠ 0: orientation of dipoles in field against thermal agitation ⇒ P = P((/T) analogous to M = M(B/T) in paramagnetism. Ferroelectric solids 8-2 Spontaneous electric polarizatin below the ferroelectric Curie temperature Ferroelectric domains and hysteresis loop P(E) analogous to ferromagnetism. Physical origin of ferroelectricity: see Fig. 8.2 Fig. 8.2 (a) The crystal structure of barium titanate. The prototype crystal is calcium titanate (perovskite). The structure is cubic, with Ba2+ ions at the cube corners, O2- ions at the face centers, and a Ti4+ ion at the body center. (b) below the Curie temperature the structure is slightly deformed, with Ba++ and Ti4+ ions displaced relative to the O2- ions, thereby developing a dipole moment. The upper and lower oxygen ions may move downward slightly. Antiferroelectric solids: analogous to antiferromagnetic solids. Local electric field 8-3 Gases: electric polarization of an atom is not affected by fields of polarized neighboring atoms. ⇒ P = ε0⋅nv α( (8.5) for identical molecules of polarizability α nv: number of molecules / volume ⇒ ε = 1 + nvα, i.e., χ = nvα (8.6) Solids: local field at an atom is influenced by dipoles of neighbours. Local field at an atom: (local = (ext + (sample P = ε0nvα(local (8.7) (8.8) Calculation of (local 8-4 a) Field EI acting on atom at the center of an imaginary sphere (I depends on crystal structure. (I = 0 if atom sites have cubic symetry, e.g. for: sc, fcc, bcc, NaCl, and CsCl structure (see practical course for derivation) b) Imaginare sphere is cut out of the solid Choise of radius R of imaginary sphere: • 2R << λ of electromagnetic field ⇒ (I = 0 for cubic symmetry. • R large enough to describe polarization in center due to atoms outside sphere by quasicontinuous dipole distribution, i.e., macroscopic polarization P ⇒ Field due to P described by polarization charges ρP on surface (see Fig. 8.3). 8-5 ρp = -Pn (8.9) Pn: normal component of P Note: unit normal points inside hollow sphere ⇒ ″-″ sign. Cf.: sphere with homogeneous surface charge density ρ ⇒ ( = (n = ρ/ε0 (Rigorous proof of eq. (8.9): see, e.g. Kittel) Fig. 8.3 In a continuum model the field at the center of a hollow sphere due to the polarization of the atoms outside the sphere can be described in termens of polarization charges on the surface of the sphere. Charge in ring at angle θ: 14 42 3 dq = − P cosθ ⋅ (2πa sin θ ) adθ Pn (8.10) Contribution to field in center: d( = − 1 dq cosθ 4πε 0 a 2 <( ⊥ P> = 0, (8.11) ( || P =( cosθ ⇒ polarization field in center of hollow sphere (″Lorentz field″) π P 1 (L = cos 2 θ sin θdθ = P 2ε 0 ∫ 3ε 0 0 8-6 (8.12) c) Account for depolarization Fig. 8.4 The depolarization field is opposite to the polarization and is due to the net surface charges originating from the polarization of a body in an external field. The polarization is homogeneous for an elipsoid of revolution. In this case the depolarizing field (N can be described in terms of a geometry-dependent depolarization factor N. Depolarizing field for homogeneous polarization (only for ellipsoid of revolution) (N =-NP / ε0 (8.13) Field inside the sample (″macroscopic field″): ( = ( ext − 1 N P, ε0 P = ε 0 χ( = N: depolarization factor 8-7 χ ε 0 ( ext 1 + Nχ (8.13a) Depolarization factor for special geometries: Sphere: N = 1/3, hollow sphere: N = -1/3 (see eq. 8.12) Long rod || ( : N = 0 ⇒ for (I = 0 : (local = (ext + (N + (L (8.14) Spherical sample: (N = -(L ⇒ (local = (ext (8.15) 1 1 2 Thin slap ⊥ ( : ( local = ( ext − ε P + 3ε P = ( ext − 3ε P (8.16) 0 0 0 1 ( local = ( ext + P Long rod || ( : (N = 0 ⇒ 3ε 0 8-8 (8.17) Clausius-Mosotti equation Solid with atom sites of cubic symetry ⇒ (I = 0 (local = ( + (L , ( = (ext + (N , (8.18) macroscopic field inside sample P = ε0nv α(local (8.8) 1 ⇒ P = ε0nv α(( + 3ε P ) 0 nvα ( 1 1 − nvα ⇒ 3 P = ε0 χ( P = ε0 ⇒ χ= (8.3) nvα 1 1 − nvα 3 ε = 1+ χ = 1+ ⇒ (8.19) (8.20) n vα 1 1 − n vα 3 ε −1 1 n vα = 3 ε +2 (8.21) Clausius-Mosotti equation (8.22) Measurement of ε ⇒ molecular polarizability α. 8-9 Lorentz’s oscillator model for the electronic polarizability Small displacement x of an electron in a lattice atom from it’s equilibrium position. ⇒ - linear restoring force - dipole moment -ex Applied electromagnetic field ⇒ ( = ( local e − iωt ⇒ forced oscillation. 0 Oscillating dipole emits adiation ⇒ damping ∝dx/dt Equation of motion: d 2x dx 2 0 m 2 + mβ + mω 0 x = −e(local e iωt dt dt (8.23) β: damping constant, ω0 = f / m : frequency of undamped vibration, Restoring force: F = -fx Analogous to forced mechanical oscillation ⇒ stationary state approached with relaxation time τ = 1 /β (8.24) 8-10 Complex amplitude: x=− 1 e ( iω local m ω2 −ω2 − 0 τ p = -ex = αelε0(local Dipole moment: e2 1 α el (ω ) = Electrical polarizability: ε 0 m ω 2 − ω 2 − iω 0 τ (8.25) (8.26) (8.27) (8.27) in (8.21) ⇒ nv e 2 ε (ω ) = 1 + ε 0m 1 1 nv e 2 = 1+ 2 ω1 e ε 0m ω 2 − ω 2 − i ω ω 02 − ω 2 − i − nv 1 τ τ 3 ε 0m (8.28) 1 e2 with ω = ω − 3 nv ε m 0 2 1 2 0 Separation: ε(ω) =ε1(ω) + iε2(ω) (8.29) nv e 2 ω12 − ω 2 ε1 (ω ) = 1 + ε 0 m (ω12 − ω 2 ) 2 + ω 2 / τ 2 (8.30) nv e 2 ω /τ ε 2 (ω ) = ε 0 m (ω12 − ω 2 ) 2 + ω 2 / τ 2 (8.31) 8-11 Maxwell equations for solid divB = 0 ⇔ ∫∫ Bd A = 0 divD = ρ ⇔ ∫∫ Dd A = Q (no magnetic monopoles) (8.32) (8.33) (charges are sources of dielectric displacement field) & curl ( = - B ⇔ & ∫ ( d r = − ∫∫ Bd A (8.34) (law of induction) & curl H = j + D ⇔ & ∫ H d r = I + ∫∫ Dd A (8.35) (current or charge in flux cause curled magnetic field) & (8.35) ⇒ curl H = j + D = σ ( + ε 0ε (& ⇒ & = − µ H = curl( ⇒ µ curl H = curl curl ( & & (8.34) ⇒ − B 0 0 − 1 curl curl ( = σ (& + ε 0ε (&& 23 µ 0 14 4 curl curl ( = ∇ × (∇ × ( ) = ∇ (∇( ) − ∇ ( 2 ∇( = 0 if ρ = 0 inside the sample ⇒ 2 ∇ ( = ε 0εµ 0(&& + µ 0σ (& Wave equation for nonmagnetic solid (µ ≈ 1) 8-12 (8.36) 2 ∇ ( = ε 0εµ 0 (&& + µ 0σ (& (8.36) Solution by damped wave ansatz: ε = ε 0e ~ i ( k ⋅ r −ω t ) ~ 2 εω 2 ω k = 2 + iσ c ε 0c 2 (8.37) in (8.37) ⇒ c = 1 / µ 0ε 0 , ⇒ (8.37) speed of light in vacuum ~ω σ ω~ k= =n ε +i c ε 0ω c ~ Complex refractive index n = n + iκ (8.38) (8.39) n: refractive index, κ: absorbtion coefficient ~ω ω k = n+i κ ⇒ c c (8.40) Plane wave propagating in z-direction: ω κω i n z −ωt z − (8.37) c c ⇒ ε = ε 0e e2 13 14 4 23 (8.40) damping in direction plane wave of propagation c in vacuum → c/n in solid 8-13 (8.41) Damping of amplitude by e Damping of intencity by e K= − − κω z c ⇒ 2κω z c 2κω : absorbtion constant c (8.42) Relation to ε(ω) =ε1(ω) + iε2(ω) ~ n 2 = ( n + iκ ) 2 = ε (ω ) = ε 1 (ω ) + iε 2 (ω ) (8.43) ε1(ω) = n2 - κ2 = ε (8.44) σ ε 0ω (8.49) ε 2 (ω ) = 2nκ = σ ~ n2 = ε +i ↑ (8.38) ⇒ ε 0ω 8-14 Fig. 8.5 General form of ε1(ω) and ε2(ω) for a dipole oscillator according to equations (8.30) and (8.31). ε2(ω) ≈ 0 outside resonance maximum Width of maximum: 1/τ ε2(ω) ≈ 0 ⇒ n ≈ ε 1 (ω ) e2 1 Resonance frequency: ω1 = ω 0 − 3 nv ε m < ω 0 0 ω0: resonance frequency of αel(ω) ω1 < ω0 due to effect of local field. Classical harmonic oscillator: ω 0 = 8-15 f /m Quantum mechnics: resonances are absorbtion frequencies due to electronic tansitions ⇒ always several resonance maxima in atom mainly in 1016 s-1 range (UV). Solid: also interband and intraband transitions Intraband transitions: between filled and empty states of conduction band in metals and semiconductors. ω < ω1 : ε1(ω) > 1 ⇒ n ≈ ε1 (ω ) > 1 lim ε 1 (ω ) = ε (0) ω →0 ε(0): statistic dielectric constant (already holds for visible range). ω > ω1 : ε1(ω) < 1 ⇒ n < 1 e.g. for X-rays lim ε1 (ω ) = 1 ω →∞ 8-16 ...
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This note was uploaded on 01/19/2010 for the course MATERIALS M504 taught by Professor Adelung during the Spring '02 term at Uni Kiel.

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