PHYS 560 Condensed Matter

Topics

  1. Free Electron Model
    1. Density of States
    2. Fermi Surface
    3. Sommerfeld Expansion
    4. Drude Model of Conductivity
  2. Lattices and Symmetries (Kittel Chapter 1 gives a good introduction)
    1. Bravais Lattice
    2. Symmetries
    3. Reciprocal lattice
  3. Non-Interacting Electrons in Periodic Potential
    1. Bloch Theorem
    2. Nearly Free Electron Model
    3. Tight Binding Model ( bra-ket vs second quantized notation )
      1. Perial Instability
      2. Wannier Function
  4. Lattice Models
    1. Dirac Hamiltonian on 2D Square Lattice
  5. Dynamics of Bloch Electrons
  6. Semiconductors
  7. Hetero-structures
  8. Phonons

Story

Condensed Matter deals with matter consists of \(10^{23}\) interacting particles. The way this huge system behaves is emergent. This is very much like the phenomenon of traffic jam. Considering a few cars on a freeway will never produce a traffic jam. Similarly, never would we understand most of the interesting phenomena in matter if we only consider a few electrons and ions. Unfortunately, we are only able to solve the exact Hamiltonian considering a few electrons and ions. Therefore approximations will have to be made.

The most widely used approximation is the single electron approximation. Within this approximation we assume that each electron moves in the same potential background without interacting with each other through electrostatic or nuclear forces. It does include the exchange interaction due to the quantum statistics of the particles.

The simplest manifestation of the single electron approximation is the free electron model. This model introduces important ideas such as density of states and Fermi surface and is (surprisingly) capable of qualitatively describing canonical conduction. The zero temperature behavior of the free Fermi gas is well described by properties of its Fermi surface, but at finite temperatures complications will arise.The canonical compromise is to use the Sommerfeld expansion.

The next step beyond the free electron model is to solve Schrodinger’s equation in a perfectly periodic potential (crystal). Naturally arising in the process are the concepts of Bravais lattice, reciprocal lattice and related symmetries. The most important observation regarding periodic potential was made by Emmanuel Bloch. He noticed that a periodic potential is discrete in Fourier space and thus only couples free electron eigenstates (plane waves) separated by a reciprocal lattice vector. This means the periodic Hamiltonian in the plane wave basis is block-diagonal thus we can solve each block separately from everything else. Practically, this means we may pick a crystal momentum and write down a rigorous form for the eigenstate of the periodic Hamiltonian. This form is known as the Bloch wave function, which is basically a linear combination of the subset of plane waves with the same crystal momentum (solving a sub-block of the full Hamiltonian).

The periodic Hamiltonian can be solved very well in the weak and strong potential limits (free electron and atoms). Two complimentary and very pedagogical approaches have been developed to solve the periodic Hamiltonian starting from the weak and strong potential limits respectively. They are the nearly free electron model and the tight binding model. Both models capture the development of bands and gaps in the energy spectrum of crystals as the potential deviations from the weak/strong limit.

Both models tell us that as we break continuous translation symmetry a gap opens up in the dispersion. Thinking about this from the other direction, the distinct points on the boundary of the Brouilln zone become degenerate as continuous translation symmetry is restored. Turns out the relationship between symmetry and degeneracy is quite profound. A robust degeneracy is usually protected by symmetry.