Berry Phases and Curvatures in Electronic-Structure Theory. David Vanderbilt Rutgers University

Transcription

1 Berry Phases and Curvatures in Electronic-Structure Theory David Vanderbilt Rutgers University

2 Rahman prize for: Theory of polarization (King-Smith & Vanderbilt) Ultrasoft pseudopotentials Three quick preliminaries: Who was Aneesur Rahman? Who is Dominic King-Smith? A parable about referee reports

3 Who was Aneesur Rahman? Father of Molecular Dynamics Born Hyderbad, India Educ. Cambridge, Louvain Argonne Natl. Labs U. Minnesota Died 1987 Rahman Prize established in 1992 with funds from IBM Photo courtesy Sam Bader via Marie-Louise Saboungi

4 Who is Dominic King-Smith? Father of Bettina PhD, Cambridge, UK Postdoc at Rutgers `91-`93 Biosym/MSI/Accelrys `93-`01 Presently at: Accelrys Job title: Product Manager, Quantum Mechanics

5 Ultrasoft Pseudopotentials

6 Berry Phases and Curvatures in Electronic-Structure Theory David Vanderbilt Rutgers University

7 Introduction By mid-1990s, density-functional perturbation theory allowed calculation of linear response to E-field However, it was not known how to: Calculate polarization itself Treat finite E-fields Analogous problem of calculating orbital magnetization also unsolved

8 Introduction Solutions of these problems are now in hand Modern theory of polarization (1993) Treatment of finite E-fields (2002) Orbital magnetization (2005) Solutions rely heavily on two crucial ingredients: Wannier functions Berry phases and related quantities This talk: Brief survey of methods! Almost nothing on applications

9 Outline of Talk Introduction Berry phases, potentials, and curvatures Realizations: Electric polarization Wannier functions Electric fields Anomalous Hall conductivity Orbital magnetization Summary and prospects

10 Berry phases u 4 Ò u 3 Ò u 2 Ò u n Ò = u 1 Ò u n-1 Ò Now take limit that density of points Æ

11 Berry phases l=1 u l Ò l=0 Continuum limit

12 (Context: Molecular coordinates) z 2 u l Ò Na 3 l=1 l=0 (z 1, z 2 ) z 1

13 Context: k-space in Brillouin zone u k Ò k y l=1 l=0 Bloch function k x 0 2p/a

14 Stokes theorem: Berry curvature u k Ò k y W k x 0 2p/a

15 Context: k-space in Brillouin zone u k Ò k y l=1 l=0 Bloch function k x 0 2p/a

16 Spanning the BZ k y l=0 l=1 u k Ò Bloch function k x 0 2p/a

17 Does any of this have any connection to real physics of materials?

18 Outline of Talk Introduction Berry phases, potentials, and curvatures Realizations: Electric polarization Wannier functions Electric fields Anomalous Hall conductivity Orbital magnetization Summary and prospects

19 P = d cell / V cell? Textbook picture (Claussius-Mossotti) But does not correspond to reality! + + +

20 Ferroelectric PbTiO 3 (Courtesy N. Marzari)

21 P = d cell / V cell? d cell =

22 P = d cell / V cell? d cell =

23 Berry-phase theory of electric polarization

24 Berry-phase theory of electric polarization Berry potential!

25 Simplify: 1 band, 1D l=0 l=1 k y u k Ò k x 0 2p/a

26 Discrete sampling of k-space

27 Discretized formula in 3D where

28 Sample Application: Born Z * +2 e? +4 e? 2 e? Paraelectric Ferroelectric 2 e?

29 Outline of Talk Introduction Berry phases, potentials, and curvatures Realizations: Electric polarization Wannier functions Electric fields Anomalous Hall conductivity Orbital magnetization Summary and prospects

30 Wannier function representation (Marzari and Vanderbilt, 1997) Wannier center

31 Mapping to Wannier centers Wannier center r n

32 Mapping to Wannier centers Wannier dipole theorem DP = S ion (Z ion e) Dr ion + S wf ( 2e) Dr wf Exact! Gives local description of dielectric response!

33 Ferroelectric BaTiO3 (Courtesy N. Marzari)

34 Wannier functions in a-si Wannier functions in l-h 2 O Fornari et al. Silvestrelli et al.

35 Wannier analysis of PVDF polymers and copolymers Courtesy S. Nakhmanson

36 Note upcoming release of public max-loc Wannier code (Organized by Nicola Marzari)

37 Outline of Talk Introduction Berry phases, potentials, and curvatures Realizations: Electric polarization Wannier functions Electric fields Anomalous Hall conductivity Orbital magnetization Summary and prospects

38 Electric Fields: The Problem Easy to do in practice: But ill-defined in principle: Zener tunneling For small E-fields, t Zener >> t Universe ; is it OK?

39 Electric Fields: The Problem y(x) is very messy is not periodic Bloch s theorem does not apply

40 Electric Fields: The Solution Seek long-lived resonance Described by Bloch functions Minimizing the electric enthalpy functional (Nunes and Gonze, 2001) Usual E KS Berry phase polarization Souza, Iniguez, and Vanderbilt, PRL 89, (2002); P. Umari and A. Pasquarello, PRL 89, (2002).

41 Electric Fields: Implementation As long as Dk is not too small: Can use standard methods to find minimum The e P term introduces coupling between k-points p/a 0 k p/a

42 Sample Application: Born Z * Can check that previous results for BaTiO 3 are reproduced

43 Outline of Talk Introduction Berry phases, potentials, and curvatures Realizations: Electric polarization Wannier functions Electric fields Anomalous Hall conductivity Orbital magnetization Summary and prospects

44 Anomalous Hall effect Ferromagnetic Material

45 Anomalous Hall effect Karplus-Luttinger theory (1954) Scattering-free, intrinsic Skew-scattering mechanism (1955) Impurity scattering Side-jump mechanism (1970) Impurity or phonon scattering Berry-phase theory (1999) Restatement of Karplus-Luttinger Semiclassical equations of motion: Sundaram and Niu, PRB 59, (1999).

46 Stokes theorem: Berry curvature u k Ò k y W k x 0 2p/a

47 Anomalous Hall conductivity of SrRuO 3 W z for k z =0 Z. Fang et al, Science 302, 92 (2003).

48 X. Wang, J. Yates, I. Souza, and D. Vanderbilt, G (Tuesday 8am). W z (k x,k z ) in bcc Fe See also Y.G. Yao et al., PRL 92, (2004).

49 Outline of Talk Introduction Berry phases, potentials, and curvatures Realizations: Electric polarization Wannier functions Electric fields Anomalous Hall conductivity Orbital magnetization Summary and prospects

50 Orbital Magnetization M is a bulk property? K fl K = M x n K is only apparently a surface property? -s +s P is a bulk property fl s = P n s is only apparently a surface property

51 Theory of orbital magnetization T. Thonhauser, D. Ceresoli, D. Vanderbilt, and R. Resta, Phys. Rev. Lett. 95, (2005). Context: Ferromagnetic insulators Single-particle approximation Vanishing magnetic field Used Wannier representation to derive a formula for the orbital magnetization

52 Orbital currents in Wannier representation Ôw s Ò Ôw s Ò Ôw s Ò vò r = + r Local Circulation (LC) Itinerant Circulation (IC)

53 T. Thonhauser, H (Tuesday 11:15am) (invited talk) Something new See also D. Xiao, J. Shi and Q. Niu, PRL 95, (2005). Berry curvature

54 Summary and Prospects Berry phases are everywhere! We discussed: Electric polarization Electric fields Anomalous Hall coefficient Orbital magnetization Other hot topics : Multiferroics and magnetoelectric effects Single graphene sheets Spin Hall effect and spin injection More Berry phases lurking around the corner?

55 Extras

56 Electric Fields: Justification Seek long-lived metastable periodic solution

57 Electric Fields: The Hitch There is a hitch! For given E-field, there is a limit on k-point sampling Length scale L C = 1/Dk Meaning: L C = supercell dimension N k = 8 L c = 8a Solution: Keep Dk > 1/L t = e/e g

58 X. Wang, J. Yates, I. Souza, and D. Vanderbilt, G (Tuesday 8am). Anomalous Hall conductivity of bcc Fe See also Y.G. Yao et al., PRL 92, (2004).

59 Orbital Magnetization K = M x n K Is M a bulk property? Is K only apparently a surface property? Definition: If K is predetermined at all surfaces in such a way that K = M x n for some vector M, then M is the bulk magnetization.

60 Orbital Magnetization Clarification: Microscopic M(r) defined via x M(r) = J(r) M(r) ill-defined: M(r) fi M(r) + M 0 + h Therefore, cannot define M as cell average of M(r) Conclusion: M is not, even in principle, a functional of the bulk current distribution J(r) (Hirst, RMP, 1997) Just as: P is not, even in principle, a functional of the bulk charge density distribution r(r)

61 Strong reasons to expect bulk M Nearsightedness: Surface current depends only on local environment Stationary quantum state: dr/dt = 0 Conservation of charge: J = 0 So: I y (A) = I y (B) = M z M z Edge of type A I y (A) I y (B) Edge of type B

62 Comparison: P vs. M Electric Polarization Defined for insulators only r(r) insufficient in principle; need access to Berry physics r operator Quantum of polarization Derivable from adiabatic theory Derivable from Wannier rep. Orbital Magnetization Insulators and metals with broken TR symmetry J(r) insufficient in principle; need access to Berry physics r v operator No quantum (no monopoles) No obvious adiabatic theory Derivable from Wannier rep.?

63 Ultrasoft Pseudopotentials Then, the good news: * Sidney Redner, Physics Today, June * (A hot paper is ) defined as a nonreview paper with 350 or more citations, an average ratio of citation age to publication age greater than two-thirds, and a citation rate increasing with time.

64 Ultrasoft Pseudopotentials Then, the good news: Sidney Redner, APS talk, March, 2004; Physics Today, June 2005.