Orbital magnetization in insulators with broken time-reversal symmetry. David Vanderbilt Rutgers University

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1 Orbital magnetization in insulators with broken time-reversal symmetry David Vanderbilt Rutgers University

2 Collaboration Collaboration Timo Thonhauser (Rutgers) David Vanderbilt (Rutgers) Davide Ceresoli (SISSA, Trieste, Italy, and Rutgers) Raffaele Resta (SISSA, Trieste, Italy) Preprints Orbital magnetization in extended systems, accepted for publication in ChemPhysChem. Orbital magnetization in periodic insulators,'

3 Motivations: Orbital Magnetization Much current interest in spintronics Magnetic semiconductors Half metallic magnets Spin injection Anomalous Hall conductivity Spin Hall effect Etc. Back to basics: H = B 4p M K surf = M n M = M spin + M orbital?

4 Motivations: Orbital Magnetization Semiclassical argument only!

5 Motivations: Orbital Magnetization Our approach: Fully quantum Based on Wannier representation Analogous to Berry-phase theory of polarization

6 Analogy: Electric Polarization P is a bulk property -s +s fl s = P n s is only apparently a surface property References King-Smith and Vanderbilt, PRB, Vanderbilt and King-Smith, PRB, Resta, RMP, Conditions and caveats: Bulk is periodic insulator Surface is Unreconstructed (1x1) Insulating, in gap common to bulk and surface s only determined up to quantum e/a surf

7 Orbital Magnetization K K = M x n 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.

8 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)

9 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

10 Central Claims of This Work Orbital magnetization is a bulk property Expandable in terms of bulk band-structure properties Closely related to Berry phases and Berry curvature Sum of two distinct contributions Suitable for calculation using standard bandstructure codes

11 Theoretical Context One-particle Hamiltonian [H,TR] 0 B macro = 0 (or commensurate) 1-particle states labeled by k Insulator Chern number C = 0 Spinless electrons 2D Isolated occupied band Tight-binding models Wannier representable For simplicity of presentation For tests

12 Vocabulary (One band in 2D)

13 Derivation of Electric Polarization

14 Derivation of Electric Polarization

15 Derivation of Orbital Magnetization? M = M LC?

16 Derivation of Orbital Magnetization??

17 Numerical Tests: Haldane model Tight-binding model of Haldane, 1988 E = +D E = D

18 Complex hoppings and flux tubes f 2f 2 f 1 t 12 = t 0 exp (+if) t 21 = t 0 exp ( if)

19 Numerical Tests: Haldane model Tight-binding model of Haldane, 1988 E = +D E = D t 2 exp(if) t 1 (real)

20 Numerical Tests: Haldane model Tight-binding model of Haldane, 1988

21 Numerical Tests: Haldane model Tight-binding model of Haldane, 1988

22 Numerical Tests: Haldane model

23 Derivation of Orbital Magnetization? M = M LC?

24 Numerical Tests: Haldane model

25 What is missing? Ôw s Ò Ôw s Ò Ôw s Ò vò r = + r Local Circulation (LC) Itinerant Circulation (IC)

26 Itinerant Circulation vò r Ôw s Ò Bulk WF: Bulk band carries no net current So vò = 0 So r vò = 0 Itinerant Circulation (IC) But what about a surface WF?

27 Numerical Tests: Haldane model Itinerant circulation does exist!

28 Numerical Tests: Haldane model

29 Numerical Tests: Haldane model Is itinerant circulation a bulk quantity?

30 Understanding Itinerant Circulation s s

31 Understanding Itinerant Circulation Region S s s Sum over blue links only

32 Understanding Itinerant Circulation Thickness: << Sample size >> Unit cell

33 Understanding Itinerant Circulation M IC can be written in terms of WFs! \ M IC is a bulk quantity!

34 Understanding Itinerant Circulation

35 Two Contributions to the Magnetization

36 Numerical Tests: Haldane model

37 Two Contributions to the Magnetization Each contribution invariant under H Æ H + DE Each contribution gauge invariant ( u k Ò Æ e ij(k) u k Ò ) Consistent with result of Xiao et al. Needed for metals or non-zero Chern

38 Future Challenges One-particle Hamiltonian [H,TR] 0 B macro = 0 (or commensurate) 1-particle states labeled by k Insulator Chern number C = 0 Spinless electrons 2D Isolated occupied band Tight-binding models Wannier representable For simplicity of presentation For tests

39 Summary Orbital magnetization is a bulk property Expandable in terms of bulk band-structure properties Closely related to Berry phases and Berry curvature Sum of two distinct contributions Suitable for calculation using standard bandstructure codes Generalizable for metals, Chern insulators?

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

Berry Phases and Curvatures in Electronic-Structure Theory David Vanderbilt Rutgers University Rahman prize for: Theory of polarization (King-Smith & Vanderbilt) Ultrasoft pseudopotentials Three quick