Solid State

Solid State

1. Crystals

   • Lattice structure consists of repeated (primitive) unit cells, some jargon to learn
     • Real space
       • Lattice vectors R = n1a1 + n2a2 + n3a3
       • Periodicity a
     • Reciprocal space
       • Reciprocal lattice vectors G = h1b1 + h2b2 + h3b3
       • Periodicity 2pi/a
   • Diffraction
     • mod(k_in) = mod(k_out)
     • Scattering wavevector q = G
     • Bragg's law and planes
     • Derivation of the "interference sum"

2. Lattice vibrations

   • Monoatomic chain
     • Collective modes and dispersion relation
       • Aliasing of wavevectors, k = k + (2pi/a)n
         • Can use the 1st Brillouin zone to describe all k's
       • Finite chain: same dispersion relation discrete set of allowed wavevectors, periodic boundary conditions
   • Diatomic chain
     • Dispersion relation with band gap, can think of it as a "perturbed" monoatomic chain
       • Acoustic mode at low energy (phonons)
       • Optic mode at high energy (couples with light)
   • 2-3D lattices
     • Can have transverse modes as well, lower phase velocity
     • Dispersion relation: sum of 1D dr for each dimension
   • Density of states and link to thermo
     • Phonons as discrete quanta of vibration (second quantisation)
       • Internal energy and heat capacity, use BE statistics without particle conservation
         • High T approximation: U proportional to T
         • Low T approximation: U proportional to T^4

3. Electron models in solids

   • Free electron model
     • Works well for metals, good first approximation for periodic potentials
     • Fermi gas - Fermi energy/wavevector/surfaces at 0 T
     • Non 0 T: Fermi-Dirac statistics
       • Quantum degenerate regime ( kbT << EF). Only electrons close to the Fermi level are affected
         • U and Cv estimate and evaluation from these mobile electrons
           • Total cv from both electrons and phonon contribution
       • Paramagnetism
         • Response of Fermi gas: Dispersion relation shifted, depending on spin orientation
           • Finite susceptibility at T = 0
   • Tight binding model
     • Single finite wells brought not to close to each other, small finite tunnelling probability
     • 2 wells in 1D, "solution" of the TISE, eigenstates
       • Hopping integral t, energy, represents the perturbation, electron in one well "sees" the other's potential
         • 2mod(t) energy splitting
     • 1D array, generalisation of 2-well case
       • Eigenstates (wave-like solutions) and energy
     • In higher dimensions, generalise previous results: dispersion relation quadratic for long wavelenghts
   • Bloch's theorem
     • Applies generally for electrons in periodic potentials, general form for eigenstates
     • Conservation of crystal momentum
   • Nearly free electron model
     • Free electron gas with periodic potential as small perturbation
       • energy gaps because of wave scattering, similar to phonon gap in diatomic chain