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
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
Berry Phase
t
In the adiabatic limit:
t    n  t  e 0
i dt  n / 

 n   d  n i
n
0

t
Geometric phase:
2
t
0
1
ei n t 
Well defined for a closed path

 n   d  n i
n

C
2
Stokes theorem
 n   d1d2 
Berry Curvature




i

 i


 1
 2
 2
 1
C
1
Analogy to Gauge Field
Berry curvature 
( )
Berry connection

 i


Geometric phase
Magnetic field

B(r )
Vector potential

A(r )
Aharonov-Bohm phase


2
 d  i     d  ( )


2
 dr A(r )   d r B(r )
Chern number
Dirac monopole

 d  ( )  integer
2

2
d
 r B(r )  integer h / e
Analogy to Riemann Geometry
Berry curvature
Gaussian curvature
Berry connection
Levi-Civita connection
Geometric phase
Holonomy angle
Chern number
Genus
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
``Berry Phase Effects on Electronic
Properties”,
by D. Xiao, M.C. Chang, Q. Niu,
Review of Modern Physics
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
Symmetry properties
• time reversal:
• space inversion:
• both:
• violation: ferromagnets, asymmetric crystals
Berry Curvature in Fe crystal
H(001)
(101)
5 105
4
4 10
3
3 10
2 102
1 101
0
0
1
-1 -10
2
-2 -10
3
-3 -10
(000)
H(100)
Anomalous Hall effect
• velocity
• distribution
g( ) = f( ) + df( )
• current
Intrinsic
Temperature dependence of AHE
J. P. Jan, Helv. Phys. Acta 25, 677 (1952)
Experiment
Mn5Ge3 : Zeng, Yao, Niu & Weitering, PRL 2006
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)
Some Magneto Effects
Density of states and specific heat:
Magnetoconductivity:
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).
Anomalous Thermoelectric Transport
• Berry phase correction
• Thermoelectric transport
Thermal Hall Conductivity
Phonon
Magnon
Electron
Wiedemann-Franz Law
Effective Quantum Mechanics
• Wavepacket energy
• Energy in canonical
variables
• Quantum theory
Spin-orbit
Spin & orbital
moment
Yafet term
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
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
Positional Shift
• Use perturbed band:
• Solve
from the first order energy correction:
• Modification of the Berry connection – the positional shift
Equations of Motion
• Effective Lagrangian:
• Equations of motion
Magnetoelectric Polarization
• Polarization in solids:
• Correction under electromagnetic fields:
• Magnetoelectric Polarization:
Magnetoelectric Polarization
Restrictions: No symmetries of time reversal, spatial inversion,
and rotation about B.
Nonlinear anomalous Hall current
• Intrinsic current:
• Results – only anomalous Hall current:
Electric-field-induced Hall Effect
• Electric field induced Hall conductivity (2-band model):
Magnetic-field induced anomalous Hall
• Model Hamiltonian:
• Magnetic field induced Hall conductivity:
• Compare to ordinary Hall effect
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