Berry Phases and Curvatures in Electronic-Structure Theory David - - PowerPoint PPT Presentation

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Berry Phases and Curvatures in Electronic-Structure Theory David - - PowerPoint PPT Presentation

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Berry Phases and Curvatures in Electronic-Structure Theory David Vanderbilt Rutgers University March APS Meeting, Baltimore, March 13 2006 Rahman prize for: Theory of polarization (King-Smith & Vanderbilt) Ultrasoft

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March APS Meeting, Baltimore, March 13 2006

Berry Phases and Curvatures in Electronic-Structure Theory

David Vanderbilt Rutgers University

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March APS Meeting, Baltimore, March 13 2006

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…
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March APS Meeting, Baltimore, March 13 2006

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”

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March APS Meeting, Baltimore, March 13 2006

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:
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March APS Meeting, Baltimore, March 13 2006

Ultrasoft Pseudopotentials

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March APS Meeting, Baltimore, March 13 2006

Berry Phases and Curvatures in Electronic-Structure Theory

David Vanderbilt Rutgers University

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March APS Meeting, Baltimore, March 13 2006

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

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March APS Meeting, Baltimore, March 13 2006

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

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March APS Meeting, Baltimore, March 13 2006

Outline of Talk

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

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

  • Summary and prospects
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March APS Meeting, Baltimore, March 13 2006

Berry phases

u3Ò u2Ò unÒ =u1Ò u4Ò … un-1Ò

Now take limit that density of points Æ∞

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March APS Meeting, Baltimore, March 13 2006

Berry phases

ulÒ

l=0 l=1 Continuum limit

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March APS Meeting, Baltimore, March 13 2006

(Context: Molecular coordinates)

ulÒ

l=0 l=1

(z1, z2)

z1 z2

Na3

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March APS Meeting, Baltimore, March 13 2006

Context: k-space in Brillouin zone

ukÒ

l=0 l=1 kx ky

2p/a

Bloch function

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March APS Meeting, Baltimore, March 13 2006

Stokes theorem: Berry curvature

ukÒ

kx ky

2p/a

W

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March APS Meeting, Baltimore, March 13 2006

Context: k-space in Brillouin zone

ukÒ

l=0 l=1 kx ky

2p/a

Bloch function

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Spanning the BZ

Bloch function

ukÒ

l=0 l=1 kx ky

2p/a

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Does any of this have any connection to real physics

  • f materials?
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March APS Meeting, Baltimore, March 13 2006

Outline of Talk

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

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

  • Summary and prospects
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March APS Meeting, Baltimore, March 13 2006

P = dcell / Vcell ?

+ – + – + – + – + – + –

  • Textbook picture

(Claussius-Mossotti)

  • But does not correspond

to reality!

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Ferroelectric PbTiO3 (Courtesy N. Marzari)

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March APS Meeting, Baltimore, March 13 2006

P = dcell / Vcell ? dcell =

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P = dcell / Vcell ? dcell =

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Berry-phase theory of electric polarization

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Berry-phase theory of electric polarization

Berry potential!

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Simplify: 1 band, 1D

ukÒ

l=0 l=1 kx ky

2p/a

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Discrete sampling of k-space

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Discretized formula in 3D

where

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March APS Meeting, Baltimore, March 13 2006

Sample Application: Born Z*

Paraelectric Ferroelectric

+2 e ? +4 e ? – 2 e ? – 2 e ?

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March APS Meeting, Baltimore, March 13 2006

Outline of Talk

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

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

  • Summary and prospects
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March APS Meeting, Baltimore, March 13 2006

Wannier function representation

(Marzari and Vanderbilt, 1997)

“Wannier center”

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Mapping to Wannier centers

Wannier center rn

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Wannier dipole theorem DP = Sion (Zione) Drion + Swf (– 2e) Drwf

  • Exact!
  • Gives local description of

dielectric response!

Mapping to Wannier centers

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Ferroelectric BaTiO3 (Courtesy N. Marzari)

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Wannier functions in a-Si

Fornari et al.

Wannier functions in l-H2O

Silvestrelli et al.

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Courtesy S. Nakhmanson

Wannier analysis of PVDF polymers and copolymers

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March APS Meeting, Baltimore, March 13 2006

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

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March APS Meeting, Baltimore, March 13 2006

Outline of Talk

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

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

  • Summary and prospects
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March APS Meeting, Baltimore, March 13 2006

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

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Electric Fields: The Problem

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

y(x) is very

messy

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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).
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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 k

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Sample Application: Born Z*

Can check that previous results for BaTiO3 are reproduced

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Outline of Talk

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

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

  • Summary and prospects
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Anomalous Hall effect

Ferromagnetic Material

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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

  • f motion:

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

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Stokes theorem: Berry curvature

ukÒ

kx ky

2p/a

W

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March APS Meeting, Baltimore, March 13 2006

  • Z. Fang et al, Science 302,

92 (2003).

Wz for kz=0

Anomalous Hall conductivity of SrRuO3

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March APS Meeting, Baltimore, March 13 2006

  • X. Wang, J. Yates, I. Souza, and D. Vanderbilt, G23.00001

(Tuesday 8am).

See also Y.G. Yao et al., PRL 92, 037204 (2004).

Wz(kx,kz) in bcc Fe

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March APS Meeting, Baltimore, March 13 2006

Outline of Talk

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

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

  • Summary and prospects
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March APS Meeting, Baltimore, March 13 2006

Orbital Magnetization

M is a bulk property?

fl fl K = M x n

K is only apparently a surface property?

K +s

  • s

P is a bulk property

fl fl s = P ⋅ n

s is only apparently a surface property

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March APS Meeting, Baltimore, March 13 2006

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

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March APS Meeting, Baltimore, March 13 2006

Orbital currents in Wannier representation r r

= +

ÔwsÒ ÔwsÒ ÔwsÒ

Local Circulation (LC) Itinerant Circulation (IC) ·vÒ

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March APS Meeting, Baltimore, March 13 2006

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).

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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?
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Extras

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Electric Fields: Justification

Seek long-lived metastable periodic solution

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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
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  • 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., PRL 92, 037204 (2004).

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Orbital Magnetization

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.

K

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Orbital Magnetization

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 functional

  • f the bulk charge density distribution r(r)

Conclusion: M is not, even in principle, a functional

  • f the bulk current distribution J(r)

(Hirst, RMP, 1997)

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Strong reasons to expect bulk M

  • Nearsightedness:

Surface current depends only

  • n local environment
  • Stationary quantum state:

dr/dt = 0

  • Conservation of charge:

—⋅J = 0

Edge of type A Edge of type B

Iy

(A)

Iy

(B)

So: Iy

(A) = Iy (B) = Mz

Mz

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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 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.?

Electric Polarization Orbital Magnetization

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March APS Meeting, Baltimore, March 13 2006

Ultrasoft Pseudopotentials

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 of citation age to publication age greater than two-thirds, and a citation rate increasing with time.”

* *

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Ultrasoft Pseudopotentials

Then, the good news:

Sidney Redner, APS talk, March, 2004; Physics Today, June 2005.