Berry phase effects on Electrons Qian Niu University
Berry phase effects on Electrons Qian Niu University of Texas at Austin Supported by DOE-NSET NSF-Focused Research Group NSF-PHY Welch Foundation International Center of Quantum Structures
Outline • Berry phase—an introduction • Bloch electron in weak fields – Anomalous velocity – Correction to phase space measure (DOS) – Apllications: AHE, orbital magnetism, etc. • Dirac electron --- degenerate bands – Orbital nature of spin – Anomalous velocity: spin orbit coupling – Incompleteness of Pauli and Luttinger Hamiltonians • Summary
Berry Phase Parameter dependent system: Adiabatic theorem: Geometric phase:
Well defined for a closed path Stokes theorem Berry Curvature
Analogies Berry curvature Magnetic field Berry connection Vector potential Geometric phase Aharonov-Bohm phase Chern number Dirac monopole
Applications • Berry phase interference, energy levels, polarization in crystals • Berry curvature spin dynamics, electron dynamics in Bloch bands • Chern number quantum Hall effect, quantum charge pump
Other Physical Effects Density of states and specific heat: Magnetoconductivity:
Electron dynamics in Dirac bands
Wave-packet in upper bands
Wave packet size Minimum size:
Mechanical observables
Zeeman energy Magnetic moment from self-rotation
Spin is a spin after all !
Wave packet dynamics
Pauli equation • Effective quantum mechanic for nonrelativistic electrons
Inconsistency between Pauli and Dirac
What is wrong with Pauli ?
Caution on effective Hamiltonians • Peierles substitution for non-degerate bands: en(k) en(p+e. A) • Luttinger Hamiltonians: – – Two-band model for conduction electrons (Rashba) Four-band model for heavy and light holes Six-band model: including spin/orbit split off Eight-band model (Kane): Zincblend semiconductors • Pauli Hamiltonian: for non-relativistic electrons • Dirac Hamiltonian: complete, or is it?
Summary Berry phase A unifying concept with many applications Bloch electron dynamics in weak fields Berry curvature: a ‘magnetic field’ in the k space. Anomalous velocity: AHE A fundamental modification of density of states Dirac electron dynamics in weak fields Orbital nature of spin Anomalous velocity: spin-orbit coupling Incompleteness of effective Hamiltonians
Acknowledgements • • • Ming-Che Chang Chih-Piao Chuu Dimitrie Culcer Ganesh Sundaram Jun-Ren Shi Di Xiao Yu-Gui Yao Chuan-Wei Zhang Ping Zhang
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