berry phase effects on bloch electrons in electromagnetic fields qian niu 牛谦 university of texas...
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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
tidti
nn
t
neett
0/
t
nnn id
0
Geometric phase:
In the adiabatic limit:
Berry Phase
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
Analogy to Gauge Field
i
)(
)(rB
)( 2
did )( )( 2 rBrdrAdr
)(rA
integer)( 2
d ehBrd r /integer )( 2
Berry curvature Gaussian curvature
Berry connection Levi-Civita connection
Geometric phase Holonomy angle
Chern number Genus
Analogy to Riemann Geometry
Applications
• Berry phaseinterference,
energy levels,
polarization in crystals
• Berry curvaturespin dynamics,
electron dynamics in Bloch bands
• Chern numberquantum 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
-103
-102
-101
0
101
102
103
104
105
-3
-2
-1
0
1
2
3
4
5
H(100)(000)
(101)H(001)
Berry Curvature in Fe crystal
Anomalous Hall effect
• velocity
• distribution g( ) = f( ) + f( )
• current
Intrinsic
J. P. Jan, Helv. Phys. Acta 25, 677 (1952)
Temperature dependence of AHE
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 EffectsDensity of states and specific heat:
Magnetoconductivity:
Orbital magnetization
Free energy
Xiao, Shi & Niu, PRL 2005Xiao, 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
Wiedemann-Franz Law
Phonon
Magnon
Electron
Effective Quantum Mechanics
• Wavepacket energy• Energy in canonical
variables
• Quantum theory
Spin-orbit
Yafet termSpin & orbital moment
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?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 ShiftPositional Shift• Use perturbed band:
• Solve from the first order energy correction:
• Modification of the Berry connection – the positional shift
Equations of MotionEquations of Motion
• Effective Lagrangian:
• Equations of motion
Magnetoelectric PolarizationMagnetoelectric Polarization
• Polarization in solids:
• Correction under electromagnetic fields:
• Magnetoelectric Polarization:
Magnetoelectric Polarization Magnetoelectric Polarization
Restrictions: No symmetries of time reversal, spatial inversion, and rotation about B.
Nonlinear anomalous Hall currentNonlinear anomalous Hall current
• Intrinsic current:
• Results – only anomalous Hall current:
Electric-field-induced Hall EffectElectric-field-induced Hall Effect
• Electric field induced Hall conductivity (2-band model):
Magnetic-field induced anomalous Magnetic-field induced anomalous HallHall
• 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