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Berry phase effects on Electrons

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

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Berry phase effects on Electrons

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  1. 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

  2. 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

  3. Berry Phase Parameter dependent system: Adiabatic theorem: Geometric phase:

  4. Well defined for a closed path Stokes theorem Berry Curvature

  5. Analogies Berry curvature Magnetic field Berry connection Vector potential Geometric phase Aharonov-Bohm phase Chern number Dirac monopole

  6. 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

  7. Other Physical Effects Density of states and specific heat: Magnetoconductivity:

  8. Electron dynamics in Dirac bands

  9. Wave-packet in upper bands

  10. Minimum size: Wave packet size

  11. Mechanical observables

  12. Magnetic moment from self-rotation Zeeman energy

  13. Spin is a spin after all !

  14. Wave packet dynamics

  15. Pauli equation • Effective quantum mechanic for non-relativistic electrons

  16. Inconsistency between Pauli and Dirac

  17. What is wrong with Pauli ?

  18. Caution on effective Hamiltonians • Peierles substitution for non-degerate bands: en(k) en(p+eA) • 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?

  19. 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

  20. Ming-Che Chang Chih-Piao Chuu Dimitrie Culcer Ganesh Sundaram Jun-Ren Shi Di Xiao Yu-Gui Yao Chuan-Wei Zhang Ping Zhang Acknowledgements

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