summary-c - Summary 4.1 Lattice vibrations Adiabatic...

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Unformatted text preview: Summary 4.1 Lattice vibrations Adiabatic approximation: decoupling of lattice vibrations and electron system (M >> me). Harmonic approximation: 1 ∂ 2U 2 U (a0 + s ) = U ( a0 ) + 2 s + .. 2 ∂r a 14 42 3 0 "spring constant " f F =− dU = − fs dS Thermal expansion requires cubic term ! Linear chain: S-9 Summary Equation of motion ⇒ dispersion relations ω(q) Accustical branch: dispersion free sound waves for q → 0. Optical branch: ω ≠ 0 for q → 0 M1 and M2 move against each other Energy gap for M1 ≠ M2 ω(q + G) = ω(q), ω(q) = ω(-q) 3D crystal: 3 accustical branches (1× longitudinal, 2× transvers) for monoatomic primitive unit cell. For each additional atom: 3 optical branches (1× longitudinal, 2× transvers) ω(q) depends on direction of q. Particle character of lattice vibrations (phonons) E.g., inelastic scattering of neutrons ( hω 0 , k 0 ) S- 0 1 Summary h k − h k 0 = ± h q + G conservation of quasimomentum hω − hω 0 = ± hω q conservation of energy 4.2 Thermal properties of crystals Periodic boundary conditions ⇒ q-quantization: qi = ni , N 1/ 3 ni = 0,1,2... q = q1g1 + q2g2 + q3g3, N: number of unit cells Debye model: elastic isotropic continuum, i.e. ω = cq ⇒ Z (ω ) = V ( 2π ) 2 1 2 3 + 3 ω 2 C L CT Quantum mechanical harmonic oscillator 1 E n = n + hω , 2 S- 1 1 n = 0,1,2,.... Summary Statistical thermodynamics < n(ω ) > T = 1 hω exp −1 kT average phonon number Specific heat of phonons 1 cv = M ∞ 1d ∂Q hωZ (ω ) < n(ω ) >T dω = ∂T V M dT ∫ 0 Heat conductivity Thermal curent density: Q = - λgradT Net flux of phonons (wave packets) from hot to cold. Q= 1 V ∑ hωα (q) < nα (q) > vα (q) α ,q cf. j = qnv (el. current density) vα = gradq ω(q) group velocity S- 2 1 Summary Harmonic approximation: no interaction between lattice vibrations ⇒ no scattering of ″hot″ phonons, λ = ∞ Anharmonic potential: phonon-phonon interaction ⇒ phonon decay and generation of new phonons ⇒ establishment of thermal equilibrium. d < n > ∂ < n > ∂ < n > = + Boltzmann equation: dt ∂t diffusion ∂t decay < n > − < n >0 ∂n = Relaxation time approximation: ∂t τ decay λ= 1 3V ∑ vα (q)Λ α (q)C α vα (q) ,q Only umklapp processes contribute to Λ. S- 3 1 Summary q1 = q2 +q3 + G ⇒ reversal of energy flux Defects, glasses Λ << Λ ideal crystal 5. Electrons in solids One-electron approximation: electron in effective field of nuclei screened by other electrons, electron-electron interaction neglected. 5.1 Free electron gas Schrödinger equation for free h particle: − 2m ∆Ψ = EΨ ⇒ h 2k 2 E (k ) = , E(k) = const. spheres 2m Ψk = Ae i k ⋅r Periodic boundary conditions: Ψ ( x + L,....) = Ψ ( x, y, z ) ⇒ 1 A= L 3/ 2 , L: finite extension of solid S- 4 1 Summary k-quantization: k x = ± 2π n x , nx= 0,1,2,... , ky =..., kz =... L Density of states: ( 2 m) 3 / 2 D( E ) = 2π 2 h 3 E − E0 E0: band edge ∞ Electron density: n = ∫ D( E ) f ( E )dE 0 f (E) = EF = 1 E − EF exp +1 kT Fermi distribution ∂F chemical potential ∂N S- 5 1 Summary Metals: f(E, T) ≈ f(E, T = 0K) Semiconductors: E −E f ( E , T ) ≈ exp F kT Boltzmann distribution generally EF − E >> 1 kT Specific heat of the electron gas ∞ ∂ 1 cV = ∫ ED( E ) f ( E )dE ρ0 ∂T T → 0: cV = γT + βT 3 π2 1 D ( E F )k 2T γ≈ 3ρ γT: electronic, βT3: phononic Electronic contribution only important for T → 0K Only electrons in small range near EF can be thermally excited due to Pauli principle. S- 6 1 Summary Thermionic emission from metals Potential well finite (typically 5eV) S- 7 1 Summary ⇒ thermionic emission possible. Work function: Φ = Evac - EF Saturation current: j s ∝ T 2 e − Φ / kT Outer field reduces Φ! 5.2 Band theory iG r ⇒ Ψk (r ) = uk (r )e i k⋅r Periodic potential of ion cores: V (r ) = ∑ VG e G Bloch waves: modulation of plane waves by lattice-periodic function uk(r). Energy gap due to reflection at Brillouin zone boundaries Necks in surfaces E(k) = const. E(k) = E(k + G) ⇒ reduction to 1st Brillouin zone possible S- 8 1 Summary reduced zone scheme S- 9 1 ...
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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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