Berry Phases and Curvatures in Electronic-Structure Theory

March APS Meeting, Baltimore, March 13 2006

Berry Phases and Curvaturesin Electronic-Structure Theory

David VanderbiltRutgers University


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…


Who was Aneesur Rahman?

Photo courtesy Sam Bader via Marie-Louise Saboungi

• !Born Hyderbad, India• Educ. Cambridge, Louvain• Argonne Natl. Labs 1960-85• U. Minnesota 1985-87• Died 1987• Rahman Prize established in

1992 with funds from IBM

“Father of Molecular Dynamics”


Who is Dominic King-Smith?

“Father of Bettina”

Accelrys Job title:“Product Manager, Quantum Mechanics”

• PhD, Cambridge, UK

• Postdoc at Rutgers `91-`93

• Biosym/MSI/Accelrys `93-`01

• Presently at:


Ultrasoft Pseudopotentials


Berry Phases and Curvaturesin Electronic-Structure Theory

David VanderbiltRutgers University


Introduction

• By mid-1990s, density-functional perturbation theoryallowed 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 magnetizationalso unsolved


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


Outline of Talk

• Introduction• Berry phases, potentials, and curvatures• Realizations:

– Electric polarization– Wannier functions– Electric fields– Anomalous Hall conductivity– Orbital magnetization

• Summary and prospects


Berry phases

u3Ò

u2Ò

unÒ =u1Ò

u4Ò

…un-1Ò

Now take limitthat density of

points Æ∞


Berry phases

ulÒ

l=0l=1

Continuumlimit


(Context: Molecular coordinates)

ulÒ

l=0l=1

(z1, z2)

z1

z2

Na3


Context: k-space in Brillouin zone

ukÒ

l=0l=1

kx

ky

0 2p/a

Bloch function


Stokes theorem: Berry curvature

ukÒ

kx

ky

0 2p/a

W


Context: k-space in Brillouin zone

ukÒ

l=0l=1

kx

ky

0 2p/a

Bloch function


Spanning the BZ

Bloch function

ukÒ

l=0 l=1

kx

ky

0 2p/a


Does any of thishave any connection

to real physicsof materials?


Outline of Talk





P = dcell / Vcell ?

+–

+–

+–

+–

+–

+–

• Textbook picture(Claussius-Mossotti)

• But does not correspondto reality!

Ferroelectric PbTiO3 (Courtesy N. Marzari)


P = dcell / Vcell ?

dcell =


Berry-phase theory of electric polarization


Berry-phase theory of electric polarization

Berry potential!


Simplify: 1 band, 1D

ukÒ

l=0 l=1

kx

ky

0 2p/a


Discrete sampling of k-space


Discretized formula in 3D

where


Sample Application: Born Z*

Paraelectric Ferroelectric

+2 e ?

+4 e ?

– 2 e ?

– 2 e ?


Outline of Talk





Wannier function representation

(Marzari and Vanderbilt,1997)

“Wannier center”


Mapping to Wannier centers

Wanniercenter

rn


Wannier dipole theorem

DP = Sion (Zione) Drion

+ Swf (– 2e) Drwf

• Exact!• Gives local description of

dielectric response!

Mapping to Wannier centers

Ferroelectric BaTiO3 (Courtesy N. Marzari)

Wannier functionsin a-Si

Fornari et al.

Wannier functionsin l-H2O

Silvestrelli et al.

Courtesy S. Nakhmanson

Wannier analysis of PVDF polymers and copolymers


Note upcoming release of public max-loc Wannier code…

(Organized by Nicola Marzari)


Outline of Talk





Electric Fields: The Problem

Easy to do in practice:

For small E-fields, tZener >> tUniverse ; is it OK?

But ill-defined in principle:Zener

tunneling


Electric Fields: The Problem

• is not periodic• Bloch’s theorem does not apply

y(x) is verymessy


Electric Fields: The Solution

• Seek long-lived resonance• Described by Bloch functions• Minimizing the electric enthalpy functional

(Nunes and Gonze, 2001)

Usual EKS

Berry phase polarization

Souza, Iniguez, and Vanderbilt, PRL 89, 117602 (2002);P. Umari and A. Pasquarello, PRL 89, 157602 (2002).


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–p/a 0k


Sample Application: Born Z*

Can check that previous resultsfor BaTiO3 are reproduced


Outline of Talk





Anomalous Hall effect

Ferromagnetic Material


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 equationsof motion:

Sundaram and Niu, PRB 59,14925 (1999).


Stokes theorem: Berry curvature

ukÒ

kx

ky

0 2p/a

W


Z. Fang et al, Science 302,92 (2003).

Wz for kz=0

Anomalous Hall conductivity of SrRuO3


X. Wang, J. Yates, I. Souza, and D. Vanderbilt, G23.00001(Tuesday 8am).

See also Y.G. Yao etal., PRL 92, 037204

(2004).

Wz(kx,kz)

in

bcc Fe


Outline of Talk





Orbital Magnetization

M is a bulk property?

fl K = M x n

K is only apparently asurface property?

K

+s-s

P is a bulk property

fl s = P ⋅ n

s is only apparently asurface property


Theory of orbital magnetization

T. Thonhauser, D. Ceresoli, D. Vanderbilt, and R. Resta,Phys. Rev. Lett. 95, 137205 (2005).

• Context:

– Ferromagnetic insulators– Single-particle approximation

– Vanishing magnetic field• Used Wannier representation to derive a

formula for the orbital magnetization


Orbital currents in Wannier representation

rr= +

ÔwsÒ ÔwsÒ ÔwsÒ

Local Circulation(LC)

Itinerant Circulation(IC)

·vÒ


Berry curvature

Something new

T. Thonhauser, H6.00001 (Tuesday 11:15am)(invited talk)

See also D. Xiao, J. Shi and Q.Niu, PRL 95, 137204 (2005).


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?


Extras


Electric Fields: Justification

Seeklong-livedmetastable

periodicsolution


Electric Fields: The Hitch

• There is a hitch!• For given E-field, there is a limit on k-point sampling• Length scale LC = 1/Dk• Meaning: LC = supercell dimension

Nk = 8

Lc = 8a

• Solution: Keep Dk > 1/Lt = e/Eg


X. Wang, J. Yates, I. Souza, and D. Vanderbilt, G23.00001(Tuesday 8am).

Anomalous Hall conductivity of bcc Fe

See also Y.G. Yao et al., PRL92, 037204 (2004).



K = M x n

Is M a bulk property?

Is K only apparently asurface property?

Definition:If K is predetermined at all surfaces in such away that K = M x n for some vector M, thenM is the bulk magnetization.

K



Clarification:

• Microscopic M(r) defined via — x M(r) = J(r)

• M(r) ill-defined: M(r) fi M(r) + M0 + —h

• Therefore, cannot define M as cell average of M(r)

Just as: P is not, even in principle, a functionalof the bulk charge density distribution r(r)

Conclusion: M is not, even in principle, a functionalof the bulk current distribution J(r)

(Hirst, RMP, 1997)


Strong reasons to expect bulk M

• Nearsightedness:Surface current depends onlyon local environment

• Stationary quantum state:dr/dt = 0

• Conservation of charge:—⋅J = 0

Edge oftype A

Edge oftype B

Iy(A)

Iy(B)

So: Iy(A) = Iy(B) = Mz

Mz


Comparison: P vs. M

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.

Insulators and metalswith 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.?

Electric Polarization Orbital Magnetization



Then, the good news:

Sidney Redner, Physics Today, June 2005.(A hot paper is…) “defined as a nonreview paper with 350 or more citations, an average ratio ofcitation age to publication age greater than two-thirds, and a citation rate increasing with time.”

*

*



Then, the good news:

Sidney Redner, APS talk,March, 2004; PhysicsToday, June 2005.

Berry Phases and Curvatures in Electronic-Structure Theory

Documents

Transcript of Berry Phases and Curvatures in Electronic-Structure Theory