Berry Phase and Anomalous Hall Effect

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Berry Phase and Anomalous Hall Effect

• Qian Niu
• University of Texas at Austin

Supported by DOE-NSET NSF-Focused Research
Group NSF-PHY Welch
Foundation
International Center of Quantum Structures
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Outline

• Berry phasean introduction
• Semiclassical transport
• Anomalous Hall effect
• Summary

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Berry Phase
Parameter dependent system
Adiabatic theorem
Geometric phase
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Well defined for a closed path
Stokes theorem
Berry Curvature
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Analogies
Berry curvature Magnetic
field Berry connection Vector
potential Geometric phase
Aharonov-Bohm phase Chern number
Dirac monopole
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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

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Hall effect

• Ordinary Hall effect (1879)
• Anomalous Hall effect (18801881)

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AHE in Fe Whisker
Slope R0
RsMs

• P.N. Dheer,
• Phys Rev (1967)

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Magnetic semiconductor

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Early theories

• Karplus Luttinger (1954)
• Intrinsic Hall conductivity
• J. Smit (1958)
• Skew scattering
• L. Berger, (1970)
• Side jump

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Our theory

• velocity
• distribution g( ) f( ) df( )
• current

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Temperature dependence of AHE
J. P. Jan, Helv. Phys. Acta 25, 677 (1952)
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Wien2000LAPW spin density functional
generalized gradient approximationspin-orbit
coupling included in the APW sphere.

Ferromagnetic bcc Fe YuguiYao et al Phys. Rev.
Lett. 92, 037204 (2004)

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Band structure bcc Fe
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Berry curvature
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Berry Curvature in the xz plane
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Comparison with experiments
Krinchik and Gushchin (1969).
Dheer (1967) 1032 (ohm cm)-1
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Kubo formula

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Transition metals anomalous Hall conductivity
(ohm cm)-1

• Theory
  Experiment
• bcc iron 750
  1030 (a)
• hcp cobalt 443 500
  (b)
• fcc nickel -2100 -753
  (c)
• (a) P.N. Dheer,Phys.Rev 156, 637 (1967).
• (b) W. Jellinghaus and M. P. De Andres, Ann.
  Phys. , 189 (1961).
• (c) J. Smit, J. Physica 21, 877 (1955).

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AHE in other systems

• MnxGa1-xAs
• Jungwirth, Niu, MacDonald Phys. Rev. Lett. 88,
  207208 (2002)
• SrRuO3
• Zhong et at,
• Science 302, 92 (2003).

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Doping dependence Jungwirth et al Appl. Phys.
Lett. 83, 320 (2003).

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AHE in two dimensionsCulcer, MacDonald,
NiuPhys. Rev. B 68 , 045327 (2003).

• Rashba

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Zincblende
n-type HgTe quantum wells X. C. Zhang et al PRB
63, 245305 (2001)
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Summary

• Berry phase
• A unifying concept with many applications
• Bloch electron dynamics in weak fields
• Berry phase around the Brillouin zone
• ------Polarization
• Berry curvature a magnetic field in the k
  space.
• -----Anomalous Hall effect
• Phase space density of states is modified.
• -----Orbital magnetization, etc.