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Semiclassical Electron dynamics: (ch. 12) o Group velocity o Effective mass: ! 1! vg = ∇ k ε " [M ] −1 αβ 1 ∂ 2ε = 2 ! ∂kα ∂k β Silicon conductionband pockets (M. Marder) o Lorentz force equation: !# !⎤ ⎡! 1 ! "k = −e ⎢ E + ∇ k ε × B ⎥ "c ⎣ ⎦ Progression to adjacent states in same band. § for sufficiently small fields § consequences: Bloch oscillations in perfect conductor Bloch oscillations & Transport § Paradox: scattering required for charge transport. § Oscillations: localized in space ~ “Lissajous figures” !" ! #k = −eE ! ! ! 1! ! ! ! v g (k ) = ∇ k ε (k ) ⇒ r (t ) = ∫ v (k (t ))dt " § Esaki idea: superlattice oscillator; “bandgap engineering” § Actual observations in semiconductor superlattices; also cooled-atom lattices. [Feldman et al. Phys Rev B 46, 7252 (1992) 1st obs.] Example of optical Bloch oscillations in artificial structure (Davoyan et al. Optics Express 2008) Showed early in semester; Fermi Transport properties in metals: surface effectively displaced by E field. More general case, over Fermi surface Electrical conductivity: Classic relationship integration with (Drude): carrier group velocity. ne τ j = −ne v = + m 2 k F = 3 3π 2 n Effective mass: large mass generally implies small σ. ne τ σ= m 2 ε F = ! 2 k F 2 2m = ! 2 (3π 2 n) 2 / 3 2m 2⎛m⎞ g (ε F ) = 2 ⎜ 2 ⎟ π ⎝! ⎠ 3/ 2 ( m 3π 2 n εF = ! 2π 2 ne 2τ 2 e 2τ g (ε F )vF2 e 2τ σ= = g (ε F )ε F = m 3 m 3 ) 1/ 3 (Expressed in terms of Fermi surface properties only) • Classical formula often applied to Fermi gas situation; scattering due to defects or phonons. or other electrons, etc. Works for spherical Fermi surface Holes and “Hole bands” § Can consider hole band as “inverted band structure”. ! ! k → −k ε → −ε −e → +e m* → − m* !# !⎤ ⎡! 1 ! "k = e ⎢ E + ∇ k ε × B ⎥ "c ⎣ ⎦ § Response of positive-charge holes equivalent to response of all the remaining filled electron states; Lorentz force and mass are reversed. § Normally useful only for small pockets. Quantum oscillations: Fermi surface measurement method, see chapter 14; de Haas-‐van Alfven measurements etc. Shubnikov de Haas effect B Closed path or open orbit ΔR Sebastian et al. Phil. Trans. A 2011 YBaCu2O3