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Phys 460 Lecture 13

( pdf version - 6 slides/page )
Monday, October 9, 2006
Lecturer: Richard Martin 		No homework assigned today

Reading:
Kittel, Chapt. 6

The Electron Gas- continued
Outline

  1. From previous lectures:
    • Typical Crystal Structures
    • Diffraction, Fourier Analysis, and the Reciprocal Lattice
    • Crystal binding, elastic waves
    • Vibration waves in crystals: dispersion curves, quantization
    • Thermal properties of crystals due to vibrations
  2. Last time:
    • Role(s) of electrons in solids
    • History: Failure of classical mechanics (Drude-Lorentz model)
    • Simplest model - non-interacting electron gas - Electrons in a box
    • Energy levels in 1 d and 3d
    • Fermi energy and momentum
    • Density of states, Internal energy, Heat capacity
    • Comparison of heat capacity of phonons and electrons in a metal
  3. Electrical Conductivity - Ohm’s Law
    • F = -eE
    • F = dp/dt = hbar dk/dt , since p = hbar k
    • All the electrons accelerate and the k points shift, i.e., the entire Fermi surface shifts
    • Electron velocity limited by scattering rate 1/tau, tau = scattering time
    • Current = j = n q v (where n = density) so that j = n q (hhar k/m) = (n q2/m) tau E
    • Thus sigma = (n q2/m) tau
    • resistivity rho = 1/sigma is additive for different mechanisms
    • Low T - defects - sigma constant; high T - phonons - sigma ~ T
  4. Hall Effect
    • Long sample: electric along length with current j flowing; perpendicular magnetic field B
    • Measure EHall in perpendicular direction
    • RHall B = EHall / (j B) = 1/(nq)
    • From usual resistivity and RHall, one can determine density n and charge of carriers
  5. Thermal Conductivity - Weidemann-Franz Law
    • Formualas just like for any gas (see phonon discussion)
    • K = (1/3) C vFermi L = (1/3) C (vFermi)2 t
    • Uisng our results for C, K = (pi2/3) (n/m) tau kB2 T
    • Justifies Weidemann-Franz Law that (K/sigma) ~ T
    • (K/sigma) = (pi2/3) (kB/e)2 T
    • Electrons dominate over phonons in good metals
    • Comparable in poor metals (alloys)
    • Phonons dominate in non-metals
  6. Metallic Binding
    • Kinetic energy always repulsive - E ~ (1/V)2/3
    • Attraction ddue to nuclei (not included in gas model - must be added)
    • Attraction energy ~ - (1/V)1/3
    • Combination leads to binding
    • Treated in Kittel only in problems
Email clarification questions and corrections to rmartin@uiuc.edu
Email questions on solving problems to xin2@.uiuc.edu