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