1. 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

2. Outline Berry phase—an introduction Semiclassical transport Anomalous Hall effect Summary

3. Geometric phase: Adiabatic theorem: Berry Phase Parameter dependent system:

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

5. Berry curvature Magnetic field Berry connectionVector potential Geometric phase Aharonov-Bohm phase Chern number Dirac monopole Analogies

6. 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

12. Hall effect Ordinary Hall effect (1879) Anomalous Hall effect (1880&1881)

13. P.N. Dheer, Phys Rev (1967) RsMsRsMs Slope = R 0 AHE in Fe Whisker

14. Magnetic semiconductor

15. Early theories Karplus & Luttinger (1954) Intrinsic Hall conductivity J. Smit (1958) –Skew scattering L. Berger, (1970) –Side jump

16. Our theory velocity distribution g( ) = f( ) +  f( ) current

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

18. Wien2000: LAPW spin density functional generalized gradient approximation spin-orbit coupling included in the APW sphere. Ferromagnetic bcc Fe YuguiYao et al: Phys. Rev. Lett. 92, 037204 (2004)

19. Band structure: bcc Fe

20. Berry curvature

21. Berry Curvature in the xz plane

22. Dheer (1967) 1032 (ohm cm)-1 Krinchik and Gushchin (1969). Comparison with experiments

23. Kubo formula

24. 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).

25. AHE in other systems Mn x Ga 1-x As: –Jungwirth, Niu, & MacDonald Phys. Rev. Lett. 88, 207208 (2002) SrRuO3: –Zhong et at, Science 302, 92 (2003).

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

27. AHE in two dimensions Culcer, MacDonald, Niu Phys. Rev. B 68, 045327 (2003). Rashba

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

29. 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.