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