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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. Outline. Berry phase—an introduction - PowerPoint PPT Presentation
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Berry phase effects on
Electrons
Qian Niu
University of Texas at Austin
Supported by DOE-NSETNSF-Focused Research GroupNSF-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
, nn
tidti
nn
t
neett
0/
t
nnn id
0
Geometric phase:
n
Adiabatic theorem:
Berry Phase
Parameter dependent system:
1
2
0
t
1 2 2 1
ii
21 ddn1
2
C
C
nnn id
Well defined for a closed path
Stokes theorem
Berry Curvature
Berry curvature Magnetic field
Berry connection Vector potential
Geometric phase Aharonov-Bohm phase
Chern number Dirac monopole
Analogies
i
)(
)(rB
)( 2
did )( )( 2 rBrdrAdr
)(rA
integer)( 2
d ehBrd r /integer )( 2
Applications
• Berry phaseinterference,
energy levels,
polarization in crystals
• Berry curvaturespin dynamics,
electron dynamics in Bloch bands
• Chern numberquantum Hall effect,
quantum charge pump
Other Physical EffectsDensity 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 non-relativistic electrons
Inconsistency between Pauli and Dirac
What is wrong with Pauli ?
Caution on effective Hamiltonians
• Peierles substitution for non-degerate bands: n(k) n(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?
Summary
Berry phaseA unifying concept with many applications
Bloch electron dynamics in weak fieldsBerry curvature: a ‘magnetic field’ in the k space.Anomalous velocity: AHEA fundamental modification of density of states
Dirac electron dynamics in weak fieldsOrbital nature of spinAnomalous velocity: spin-orbit couplingIncompleteness 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