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Polarization, electric fields, and dielectric response in insulators - - PowerPoint PPT Presentation

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Polarization, electric fields, and dielectric response in insulators David Vanderbilt Rutgers University Conference on Computational Physics, Los Angeles, 2005 http://www.physics.rutgers.edu/~dhv/talks/ccp05.pdf Outline Introduction

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SLIDE 1

Conference on Computational Physics, Los Angeles, 2005

http://www.physics.rutgers.edu/~dhv/talks/ccp05.pdf

Polarization, electric fields, and dielectric response in insulators

David Vanderbilt Rutgers University

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SLIDE 2

Conference on Computational Physics, Los Angeles, 2005

http://www.physics.rutgers.edu/~dhv/talks/ccp05.pdf

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
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Conference on Computational Physics, Los Angeles, 2005

http://www.physics.rutgers.edu/~dhv/talks/ccp05.pdf

Principal Contributors:

  • D. King-Smith Polarization
  • N. Marzari Wannier functions
  • R. Nunes
  • I. Souza
  • J. Iniguez
  • N. Sai
  • O. Dieguez
  • K. Rabe
  • X. Wu
  • D. Hamann
  • X. Wang DFPT in presence of E-field

Collaborators

Electric fields Mapping E vs. P Systematic DFPT in E and strain

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SLIDE 4

Conference on Computational Physics, Los Angeles, 2005

http://www.physics.rutgers.edu/~dhv/talks/ccp05.pdf

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).

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SLIDE 5

Conference on Computational Physics, Los Angeles, 2005

http://www.physics.rutgers.edu/~dhv/talks/ccp05.pdf

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

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SLIDE 6

Conference on Computational Physics, Los Angeles, 2005

http://www.physics.rutgers.edu/~dhv/talks/ccp05.pdf

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 on applications

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SLIDE 7

Conference on Computational Physics, Los Angeles, 2005

http://www.physics.rutgers.edu/~dhv/talks/ccp05.pdf

  • 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

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SLIDE 8

Conference on Computational Physics, Los Angeles, 2005

http://www.physics.rutgers.edu/~dhv/talks/ccp05.pdf

P = dsample / Vsample ?

+s

  • s

DP = ( L2 s ) . L / L3 L x L x L sample:

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SLIDE 9

Conference on Computational Physics, Los Angeles, 2005

http://www.physics.rutgers.edu/~dhv/talks/ccp05.pdf

  • 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

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SLIDE 10

Conference on Computational Physics, Los Angeles, 2005

http://www.physics.rutgers.edu/~dhv/talks/ccp05.pdf

P = dcell / Vcell ?

+ – + – + – + – + – + –

  • Textbook picture

(Claussius-Mossotti)

  • But does not correspond

to reality!

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SLIDE 11

Ferroelectric PbTiO3 (Courtesy N. Marzari)

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Conference on Computational Physics, Los Angeles, 2005

http://www.physics.rutgers.edu/~dhv/talks/ccp05.pdf

P = dcell / Vcell ? dcell = 0

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Conference on Computational Physics, Los Angeles, 2005

http://www.physics.rutgers.edu/~dhv/talks/ccp05.pdf

P = dcell / Vcell ? dcell =

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SLIDE 14

Conference on Computational Physics, Los Angeles, 2005

http://www.physics.rutgers.edu/~dhv/talks/ccp05.pdf

P = dcell / Vcell ? dcell =

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SLIDE 15

Conference on Computational Physics, Los Angeles, 2005

http://www.physics.rutgers.edu/~dhv/talks/ccp05.pdf

  • 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

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SLIDE 16

Conference on Computational Physics, Los Angeles, 2005

http://www.physics.rutgers.edu/~dhv/talks/ccp05.pdf

  • 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

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SLIDE 17

Conference on Computational Physics, Los Angeles, 2005

http://www.physics.rutgers.edu/~dhv/talks/ccp05.pdf

  • 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 ? d) P µ Snk ·unk˙i—k˙unkÒ ?

Theory of electric polarization

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Conference on Computational Physics, Los Angeles, 2005

http://www.physics.rutgers.edu/~dhv/talks/ccp05.pdf

Attempt 4

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Conference on Computational Physics, Los Angeles, 2005

http://www.physics.rutgers.edu/~dhv/talks/ccp05.pdf

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Conference on Computational Physics, Los Angeles, 2005

http://www.physics.rutgers.edu/~dhv/talks/ccp05.pdf

Simplify: 1 band, 1D

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Conference on Computational Physics, Los Angeles, 2005

http://www.physics.rutgers.edu/~dhv/talks/ccp05.pdf

Discrete sampling of k-space

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Conference on Computational Physics, Los Angeles, 2005

http://www.physics.rutgers.edu/~dhv/talks/ccp05.pdf

Gauge invariance

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Conference on Computational Physics, Los Angeles, 2005

http://www.physics.rutgers.edu/~dhv/talks/ccp05.pdf

Discretized formula in 3D

where

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Conference on Computational Physics, Los Angeles, 2005

http://www.physics.rutgers.edu/~dhv/talks/ccp05.pdf

Sample Application: Born Z*

Paraelectric Ferroelectric

+2 e ? +4 e ? – 2 e ? – 2 e ?

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SLIDE 25

Conference on Computational Physics, Los Angeles, 2005

http://www.physics.rutgers.edu/~dhv/talks/ccp05.pdf

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

– Mapping energy vs. polarization – Systematic treatment of E-fields and strains

  • Summary and prospects
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SLIDE 26

Conference on Computational Physics, Los Angeles, 2005

http://www.physics.rutgers.edu/~dhv/talks/ccp05.pdf

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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SLIDE 27

Conference on Computational Physics, Los Angeles, 2005

http://www.physics.rutgers.edu/~dhv/talks/ccp05.pdf

Electric Fields: The Problem

  • is not periodic
  • Bloch’s theorem does not apply
  • acts as singular perturbation
  • n eigenfunctions
  • not bounded from below
  • There is no ground state

y(x) is very

messy

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SLIDE 28

Conference on Computational Physics, Los Angeles, 2005

http://www.physics.rutgers.edu/~dhv/talks/ccp05.pdf

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?
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SLIDE 29

Conference on Computational Physics, Los Angeles, 2005

http://www.physics.rutgers.edu/~dhv/talks/ccp05.pdf

Electric Fields: Justification

Seek long-lived metastable periodic solution

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SLIDE 30

Conference on Computational Physics, Los Angeles, 2005

http://www.physics.rutgers.edu/~dhv/talks/ccp05.pdf

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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SLIDE 31

Conference on Computational Physics, Los Angeles, 2005

http://www.physics.rutgers.edu/~dhv/talks/ccp05.pdf

Sample Application: Born Z*

Can check that previous results for BaTiO3 are reproduced

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SLIDE 32

Conference on Computational Physics, Los Angeles, 2005

http://www.physics.rutgers.edu/~dhv/talks/ccp05.pdf

Sample Application: Born Z*

(Souza,Iniguez, and Vanderbilt, 2002)

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SLIDE 33

Conference on Computational Physics, Los Angeles, 2005

http://www.physics.rutgers.edu/~dhv/talks/ccp05.pdf

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

– Mapping energy vs. polarization – Systematic treatment of E-fields and strains

  • Summary and prospects
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SLIDE 34

Conference on Computational Physics, Los Angeles, 2005

http://www.physics.rutgers.edu/~dhv/talks/ccp05.pdf

Wannier function representation

(Marzari and Vanderbilt, 1997)

“Wannier center”

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Conference on Computational Physics, Los Angeles, 2005

http://www.physics.rutgers.edu/~dhv/talks/ccp05.pdf

Mapping to Wannier centers

Wannier center rn

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Conference on Computational Physics, Los Angeles, 2005

http://www.physics.rutgers.edu/~dhv/talks/ccp05.pdf

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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SLIDE 37

Ferroelectric BaTiO3 (Courtesy N. Marzari)

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SLIDE 38

Wannier functions in a-Si

Fornari et al.

Wannier functions in l-H2O

Silvestrelli et al.

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SLIDE 39
  • S. Nakhmanson et al. (W26.3 2:54pm Thursday)

Wannier analysis of PVDF polymers and copolymers

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SLIDE 40

Conference on Computational Physics, Los Angeles, 2005

http://www.physics.rutgers.edu/~dhv/talks/ccp05.pdf

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
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SLIDE 41

Conference on Computational Physics, Los Angeles, 2005

http://www.physics.rutgers.edu/~dhv/talks/ccp05.pdf

Systematic treatment of E-fields and strain

We identify six needed elementary tensors:

tensor ric piezoelect ion

  • Frozen

sor strain ten Internal tensor charge effective Dynamical matrix constant

  • Force

tnsor elastic ion

  • Frozen

tensor dielectric ion

  • Frozen

= = L = = = =

j mj m mn jk

e Z K C

a a ab

c

These are computed within ABINIT using DFPT methods. What are they?

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

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SLIDE 42

Conference on Computational Physics, Los Angeles, 2005

http://www.physics.rutgers.edu/~dhv/talks/ccp05.pdf

  • field

E Strain Displacement

  • L
  • L

c

  • e
  • Z
  • e
  • Z

K C

  • field

E Strain Displacement

They are elements of “big Hessian matrix”

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SLIDE 43

Conference on Computational Physics, Los Angeles, 2005

http://www.physics.rutgers.edu/~dhv/talks/ccp05.pdf

Elementary Tensors Build from Relaxed-ion tensors To

j mj m jk mn

e Z C K

a a ab

c L

j jk

e C

a ab

c

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Conference on Computational Physics, Los Angeles, 2005

http://www.physics.rutgers.edu/~dhv/talks/ccp05.pdf ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) (

) ( ) ( ) ( , ,

e s aa a a b h ab a b s ab a a e a h h b a ab s ab b h ab a e a ab

s b b e b c c b c

jj j j j j j j k jk j jk jk k jk j k j jk D jk

S d k e h d g e S d C S e C e e e C C = = = = = = + = + =

  • 1

1 1 j jk

compute to e C ion relaxed Use

Elastic tensor at fixed D Free-stress dielectric tensor Elastic compliance tensor Inverse dielectric tensor Various piezoelectric tensors Electromechanical coupling

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SLIDE 45

Conference on Computational Physics, Los Angeles, 2005

http://www.physics.rutgers.edu/~dhv/talks/ccp05.pdf

Metallic Metallic Short circuit boundary condition Apply strain perturbation Measuring stress response and get C

C(

(e e) )

44 43 43 260 114 114 114 231 144 114 144 231

Metallic Metallic Open circuit boundary condition Apply strain perturbation Measuring stress response and get C

C(

(D) D)

C C(

(D) D) (

(GPa GPa) )

44 40 40 242 123 123 123 226 139 123 139 226

C C(

(e

e)

) (

(GPa GPa) )

Elastic tensors at different elec. BC’s: ZnO

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SLIDE 46

Conference on Computational Physics, Los Angeles, 2005

http://www.physics.rutgers.edu/~dhv/talks/ccp05.pdf

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
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SLIDE 47

Conference on Computational Physics, Los Angeles, 2005

http://www.physics.rutgers.edu/~dhv/talks/ccp05.pdf

Mapping Energy vs. Polarization

BaTiO3

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

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SLIDE 48

Conference on Computational Physics, Los Angeles, 2005

http://www.physics.rutgers.edu/~dhv/talks/ccp05.pdf

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)

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SLIDE 49

Conference on Computational Physics, Los Angeles, 2005

http://www.physics.rutgers.edu/~dhv/talks/ccp05.pdf

Summary and Prospects

  • Electric polarization

– Problem and solution

  • Electric fields

– Problem and solution

  • Localized description:

– Wannier functions

  • Dielectric and piezoelectric properties

– Mapping energy vs. polarization – Systematic treatment of E-fields and strains

  • 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