1. Topology of Complex Energy Bands in the Resonant Scattering Problem An exercise in complex variable theory applied to condensed matter physics

2. Resonant Scattering If a defect tends to form a bound state at energy E b, then propagating states close to this energy are very strongly scattered e.g. GaAs with a small concentration of N replacing some As atoms EbEb Modified Band Structure

3. Momentum States Current-carrying states scattering by defects momentum is not exactly conserved momentum states are not exact energy eigenstates best approximation to an eigenstate with momentum k : Lifetime = Energy of momentum state k is shifted by scattering from E k to z = E - iγ

4. Self-consistent Greens Function Defect Greens function : Momentum Greens function : Defect energy broadening: Solve Self- consistently 4 Greens Function: G(z) diverges at excitation energies ΔE is complex

5. The Poles of Momentum Greens Function The poles Greens function The poles of G kk occur at : Band energies z are then defined by: 5 Defines continuous curves z(ε) in the complex plane Project: investigate the topology of these curves

6. Project Challenges Understanding Green's functions in this context Understanding complex analytic function theory associated with the Green's function Developing a numerical approach to solve the complex energy equation (programming & solution) Interpreting the complex bands to give physical quantities: density of states, group velocity, etc.