1. Berry Phase Effects on Bloch Electrons in Electromagnetic Fields
Qian Niu 牛谦
University of Texas at Austin 北京大学
Collaborators:
Y. Gao, Shengyuan Yang, C.P. Chuu, D. Xiao, W. Yao,
D. Culcer, J.R.Shi, Y.G. Yao, G. Sundaram, M.C. Chang,
T. Jungwirth, J. Sinova, A.H.MacDonald
H. Weitering, J. Beach, M. Tsoi, J. Erskine
Supported by : DOE, NSF, Welch Foundation

2. Outline Berry phase and its applications
Berry curvature in momentum space
Anomalous velocity, Anomalous Hall effect
Density of States, Orbital Magnetization
Effective Quantum Mechanics
Second order extension
Positional shift
Magnetoelectric Polarization
Nonlinear anomalous Hall effect

3. Berry Phase
In the adiabatic limit:
Geometric phase:

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

5. Analogy to Gauge Field Berry curvature Magnetic field
Berry connection Vector potential
Geometric phase Aharonov-Bohm phase
Chern number Dirac monopole

6. Analogy to Riemann Geometry
Berry curvature Gaussian curvature
Berry connection Levi-Civita connection
Geometric phase Holonomy angle
Chern number Genus

7. Applications Berry phase Berry curvature Chern number interference,
energy levels,
polarization in crystals
Berry curvature
spin dynamics,
electron dynamics in Bloch bands
Chern number
quantum Hall effect,
quantum charge pump
``Berry Phase Effects on Electronic Properties”,
by D. Xiao, M.C. Chang, Q. Niu,
Review of Modern Physics

8. Outline Berry phase and its applications
Berry curvature in momentum space
Anomalous velocity, Anomalous Hall effect
Density of States, Orbital Magnetization
Effective Quantum Mechanics
Second order extension
Positional shift
Magnetoelectric Polarization
Nonlinear anomalous Hall effect

10. Symmetry properties time reversal: space inversion: both:
violation: ferromagnets, asymmetric crystals

11. Berry Curvature in Fe crystal

12. Anomalous Hall effect velocity distribution g( ) = f( ) + df( )
current
Intrinsic

13. Temperature dependence of AHE
J. P. Jan, Helv. Phys. Acta 25, 677 (1952)

14. Experiment Mn5Ge3 : Zeng, Yao, Niu & Weitering, PRL 2006

15. Intrinsic AHE in other ferromagnets
Semiconductors, MnxGa1-xAs
Jungwirth, Niu, MacDonald , PRL (2002) , J Shi’s group (2008)
Oxides, SrRuO3
Fang et al, Science , (2003).
Transition metals, Fe
Yao et al, PRL (2004),Wang et al, PRB (2006), X.F. Jin’s group (2008)
Spinel, CuCr2Se4-xBrx
Lee et al, Science, (2004)
First-Principle Calculations-Review
Gradhand et al (2012)

17. Some Magneto Effects Density of states and specific heat:
Magnetoconductivity:

18. Orbital magnetization
Free energy
Xiao, Shi & Niu, PRL 2005
Xiao, et al, PRL 2006
Also see:
Thonhauser et al, PRL (2005).
Ceresoli et al PRB (2006).

19. Anomalous Thermoelectric Transport
Berry phase correction
Thermoelectric transport

20. Thermal Hall Conductivity
Phonon
Magnon
Electron
Wiedemann-Franz Law

21. Effective Quantum Mechanics
Wavepacket energy
Energy in canonical variables
Quantum theory
Spin-orbit
Spin & orbital moment
Yafet term

22. Outline Berry phase and its applications
Berry curvature in momentum space
Anomalous velocity, Anomalous Hall effect
Density of States, Orbital magnetization
Effective Quantum Mechanics
Second order extension
Positional Shift
Magnetoelectric Polarization
Nonlinear anomalous Hall effect

23. Why second order? In many systems, first order response vanishes
May be interested in magnetic susceptibility, electric polarizebility, magneto-electric coupling
Magnetoresistivity
Nonlinear anomalous Hall effects

24. Positional Shift Use perturbed band:
Solve from the first order energy correction:
Modification of the Berry connection – the positional shift

25. Equations of Motion
Effective Lagrangian:
Equations of motion

26. Magnetoelectric Polarization
Polarization in solids:
Correction under electromagnetic fields:
Magnetoelectric Polarization:

27. Magnetoelectric Polarization
Restrictions: No symmetries of time reversal, spatial inversion, and rotation about B.

28. Nonlinear anomalous Hall current
Intrinsic current:
Results – only anomalous Hall current:

29. Electric-field-induced Hall Effect
Electric field induced Hall conductivity (2-band model):

30. Magnetic-field induced anomalous Hall
Model Hamiltonian:
Magnetic field induced Hall conductivity:
Compare to ordinary Hall effect

31. Conclusion Berry phase is a unifying concept
Berry curvature in momentum space
Anomalous velocities, Anomalous Hall effect
Density of states, Orbital magnetization
Effective quantum Mechanics
Second order extension
Positional shift
Magnetoelectric polarization
Nonlinear anomalous Hall effect