Berry phase effects on Electrons

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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
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Outline

• Berry phasean 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

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Berry Phase
Parameter dependent system
Adiabatic theorem
Geometric phase
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Well defined for a closed path
Stokes theorem
Berry Curvature
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Analogies
Berry curvature Magnetic
field Berry connection Vector
potential Geometric phase
Aharonov-Bohm phase Chern number
Dirac monopole
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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

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Other Physical Effects

• Density of states and specific heat

Magnetoconductivity
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Electron dynamics in Dirac bands
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Wave-packet in upper bands
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Wave packet size
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Mechanical observables
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Zeeman energy
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Spin is a spin after all !
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Wave packet dynamics
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Pauli equation

• Effective quantum mechanic for non-relativistic
  electrons

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Inconsistency between Pauli and Dirac
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What is wrong with Pauli ?
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Caution on effective Hamiltonians

• Peierles substitution for non-degerate bands
• en(k)
  en(peA)
• 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?

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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

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Acknowledgements

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