Topology of Complex Energy Bands in the Resonant Scattering Problem

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 Eb, then propagating states close to this energy are very strongly scattered • e.g. GaAs with a small concentration of N replacing some As atoms Modified Band Structure Eb

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 Ek to z = E - iγ

4. Self-consistent Green’s Function Green’s Function: G(z) diverges at excitation energies • MomentumGreen’s function: Solve Self- consistently • Defect energy broadening: ΔE is complex • DefectGreen’s function:

5. The Poles of Momentum Green’s Function • The polesGreen’s function • The poles of Gkk occur at : • Band energies z are then defined by: 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.