polarization integration

Conference on Computational Physics, Los Angeles, 2005http://www.physics.rutgers.edu/~dhv/talks/ccp05.pdf

Polarization, electric fields, anddielectric response in insulators

David VanderbiltRutgers University


Outline

• Introduction• Electric polarization

– What is the problem?– What is the solution?

• Electric fields– What is the problem?– What is the solution?

• Localized description:– Wannier functions

• Dielectric and piezoelectric properties– Systematic treatment of E-fields and strains– Mapping energy vs. polarization

• Summary and prospects


Principal Contributors:D. King-Smith PolarizationN. Marzari Wannier functionsR. NunesI. SouzaJ. IniguezN. SaiO. DieguezK. RabeX. WuD. HamannX. Wang DFPT in presence of E-field

Collaborators

Electric fields

Mapping E vs. P

Systematic DFPT in E and strain


Principal References

• Polarization– R.D. King-Smith and D. Vanderbilt, Phys. Rev. B 47, 1651 (1993).

• Review on polarization– R. Resta, Rev. Mod. Phys. 66, 899 (1994).

• Dynamics of polarization– I. Souza, J. Iniguez and D. Vanderbilt, Phys. Rev. B 69, 085106 (2004).

• Finite electric field– R.W. Nunes and X. Gonze, Phys. Rev. B 63, 155107 (2001).– I. Souza, J. Iniguez and D. Vanderbilt, Phys. Rev. Lett. 89, 117602 (2002).– P. Umari and A. Pasquarello, Phys. Rev. Lett. 89, 157602 (2002).

• DFPT in E-field– X. Wang and D. Vanderbilt, in preparation.

• Mapping energy vs. polarization– N. Sai, K.M. Rabe, and D. Vanderbilt, Phys. Rev. B 66, 104108 (2002).– O. Dieguez and D. Vanderbilt, in preparation.

• Systematic DFPT for E-fields and strain– X. Wu, D. Vanderbilt, and D.R. Hamann, submitted to Physical Review B.– D.R. Hamann, X. Wu, K.M. Rabe, and D. Vanderbilt, and, Phys. Rev. B. 71, 035117 (2005).


Introduction

• Context: DFT (density functional theory)• By mid-1990s, linear-response (DFPT)

allowed calculation of:– Response of P to any perturbation– Response of anything to E-field perturbation

• However, it was not known how to:– Calculate P itself– Treat finite E-fields


Introduction

• Solutions of these problems are now in hand– Modern theory of polarization (1993)– Treatment of finite E-fields (2002)

• Allows routine calculation of non-linear dielectric,piezoelectric properties of complex materials

This talk:

Emphasis is on methods!

Touch only very briefly onapplications


• Electric polarization:P = d / volume

• How to define as a bulk quantity?a) P = dsample / Vsample ?b) P = dcell / Vcell ?c) P µ Snk ·ynk˙r˙ynkÒ ?

Theory of electric polarization


P = dsample / Vsample ?

+s-s

DP = ( L2 s ) . L / L3

L x L x L sample:


P = dcell / Vcell ?

+–

+–

+–

+–

+–

+–

• Textbook picture(Claussius-Mossotti)

• But does not correspondto reality!

Ferroelectric PbTiO3 (Courtesy N. Marzari)


P = dcell / Vcell ?

dcell = 0


P = dcell / Vcell ?

dcell =



• How to define as a bulk quantity?a) P = dsample / Vsample ?b) P = dcell / Vcell ?c) P µ Snk ·ynk˙r˙ynkÒ ?d) P µ Snk ·unk˙i—k˙unkÒ ?



Attempt 4


Simplify: 1 band, 1D


Discrete sampling of k-space


Gauge invariance


Discretized formula in 3D

where


Sample Application: Born Z*

Paraelectric Ferroelectric

+2 e ?

+4 e ?

– 2 e ?

– 2 e ?


Outline





• Dielectric and piezoelectric properties– Mapping energy vs. polarization– Systematic treatment of E-fields and strains



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• acts as singular perturbation

on eigenfunctions• not bounded from below• There is no ground state

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

• Justification?


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



Can check that previous resultsfor BaTiO3 are reproduced



(Souza,Iniguez,and Vanderbilt,

2002)


Outline








Wannier function representation

(Marzari andVanderbilt, 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.

S. Nakhmanson et al. (W26.3 2:54pm Thursday)

Wannier analysis of PVDF polymers and copolymers


Outline








Systematic treatment of E-fields and strain

We identify six needed elementary tensors:

tensorricpiezoelection -Frozen

sorstrain ten Internal tensorcharge effective Dynamical

matrixconstant -Force

tnsorelasticion -Frozen

tensordielectricion -Frozen

=

=L==

=

=

j

mj

m

mn

jk

e

ZK

C

a

a

abc

These are computed within ABINIT using DFPT methods.

What are they?

(X. Wu, D. Vanderbilt, and D.R. Hamann, submitted to PRB)


- fieldEStrainDisplacement

-L

-L

c

-e-Z

-e

-ZK

C

- fieldE

Strain

Displacement

They are elements of “big Hessian matrix”


Elementary Tensors

Build from

Relaxed-ion tensors

To

j

mj

m

jk

mn

e

Z

C

K

a

a

abc

L j

jk

e

C

a

abc


)()(

)(

)(

)(

)()()(

)(

)()()(

)(

)(

)(

,,

es

aa

a

a

b

h

aba

b

s

aba

a

e

a

hh

baab

s

ab

b

h

aba

e

aab

s

b

b

eb

cc

b

c

jj

j

j

jj

jj

kjkj

jkjk

kjkj

kjjk

D

jk

S

dk

eh

dg

eSd

CS

eCe

eeCC

=

=

=

=

=

=

+=

+=

-

-

-

1

1

1

jjkcompute toe C ion relaxed Use

Elastic tensor at fixed D

Free-stress dielectric tensor

Elastic compliance tensor

Inverse dielectric tensor

Various piezoelectric tensors

Electromechanical coupling


Metallic

Metallic

Short circuit boundary condition

Apply strain perturbation

Measuring stress response and get CC((ee))

4400000

0430000

0043000

000260114114

000114231144

000114144231Metallic

Metallic

Open circuit boundary condition

Apply strain perturbation

Measuring stress response and get CC((D)D)

CC((D) D) ((GPaGPa))

4400000

0400000

0040000

000242123123

000123226139

000123139226

CC((ee)) ((GPaGPa))

Elastic tensors at different elec. BC’s: ZnO


Outline








Mapping Energy vs. Polarization

BaTiO3 (Courtesy N. Marzari)

Oswaldo Dieguez (W26.7 3:42pm Thursday)


Status of Implementation in Code Packages

• Electric polarization– All major codes: ABINIT, PWSCF, VASP, CPMD, SIESTA, CRYSTAL, etc.

• Electric fields– ABINIT (courtesy I. Souza, J. Iniguez, M. Veithen)

• Maximally localized Wannier functions:– Package at www.wannier.org (courtesy N. Marzari)

• Systematic treatment of E-fields and strains– ABINIT (courtesy X. Wu, D.R. Hamann, K. Rabe)

• DFPT in finite electric field– Coming to ABINIT soon (courtesy X. Wang)

• Mapping energy vs. P– Coming to ABINIT soon (courtesy O. Dieguez)


Summary and Prospects

• Electric polarization– Problem and solution

• Electric fields– Problem and solution



• New directions:– Dynamic generalizations of Pberry

(I. Souza, Valley Prize Talk, B3.1 11:15am Monday)– DFPT in finite electric field

(X. Wang, S32.3 2:30pm Wednesday)• Many possible applications

polarization integration

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