Polarization, electric fields, and dielectric response in insulators

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Conference on Computational Physics, Los Angeles, 2005 http://www.physics. rutgers.edu/~dhv/talks/ccp05.pdf. Polarization, electric fields, and dielectric ...
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 • 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 Conference on Computational Physics, Los Angeles, 2005 http://www.physics.rutgers.edu/~dhv/talks/ccp05.pdf

Collaborators Principal Contributors: D. King-Smith N. Marzari R. Nunes I. Souza J. Iniguez N. Sai O. Dieguez K. Rabe X. Wu D. Hamann X. Wang

Polarization Wannier functions Electric fields

Mapping E vs. P Systematic DFPT in E and strain DFPT in presence of E-field

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

Principal References •

Polarization –



Review on polarization –



X. Wang and D. Vanderbilt, in preparation.

Mapping energy vs. polarization – –



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 –



I. Souza, J. Iniguez and D. Vanderbilt, Phys. Rev. B 69, 085106 (2004).

Finite electric field – – –



R. Resta, Rev. Mod. Phys. 66, 899 (1994).

Dynamics of polarization –



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

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).

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

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

Theory of electric polarization • 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 ? Conference on Computational Physics, Los Angeles, 2005 http://www.physics.rutgers.edu/~dhv/talks/ccp05.pdf

P = dsample / Vsample ? L x L x L sample:

-s

+s DP = ( L2 s ) . L / L3

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

Theory of electric polarization • 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 ? 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!

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

Ferroelectric PbTiO3

(Courtesy N. Marzari)

P = dcell / Vcell ?

dcell = 0

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

P = dcell / Vcell ?

dcell =

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

P = dcell / Vcell ?

dcell =

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

Theory of electric polarization • 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 ? Conference on Computational Physics, Los Angeles, 2005 http://www.physics.rutgers.edu/~dhv/talks/ccp05.pdf

Theory of electric polarization • 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 ? Conference on Computational Physics, Los Angeles, 2005 http://www.physics.rutgers.edu/~dhv/talks/ccp05.pdf

Theory of electric polarization • Electric polarization: P = d / volume • How to define as a bulk quantity? a) b) c) d)

P P P P

= dsample / Vsample ? = dcell / Vcell ? µ Snk ·ynk˙r˙ynkÒ ? µ Snk ·unk˙i—k˙unkÒ ? Conference on Computational Physics, Los Angeles, 2005 http://www.physics.rutgers.edu/~dhv/talks/ccp05.pdf

Attempt 4

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

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

Simplify: 1 band, 1D

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

Discrete sampling of k-space

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

Gauge invariance

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

Discretized formula in 3D

where

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

Sample Application: Born Z*

+2 e ? +4 e ? –2e ? Paraelectric

Ferroelectric

–2e ?

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 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:

But ill-defined in principle:

Zener tunneling

For small E-fields, tZener >> tUniverse ; is it OK? Conference on Computational Physics, Los Angeles, 2005 http://www.physics.rutgers.edu/~dhv/talks/ccp05.pdf

Electric Fields: The Problem

y(x) is very messy

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

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? 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

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 L c = 8a

• Solution: Keep Dk > 1/Lt =

e/Eg

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 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)

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

Mapping to Wannier centers

Wannier center rn

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

Mapping to Wannier centers Wannier dipole theorem

DP =

Sion (Zione) Drion + Swf (– 2e) Drwf

• Exact! • Gives local description of dielectric response! Conference on Computational Physics, Los Angeles, 2005 http://www.physics.rutgers.edu/~dhv/talks/ccp05.pdf

Ferroelectric BaTiO3

(Courtesy N. Marzari)

Wannier functions in a-Si

Wannier functions in l-H2O

Fornari et al.

Silvestrelli et al.

Wannier analysis of PVDF polymers and copolymers

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

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

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

Systematic treatment of E-fields and strain (X. Wu, D. Vanderbilt, and D.R. Hamann, submitted to PRB)

We identify six needed elementary tensors: cab = Frozen - ion dielectric tensor C jk = Frozen - ion elastic tnsor K mn = Force - constant matrix Z ma = Dynamical effective charge tensor L mj = Internal strain tensor eaj = Frozen - ion piezoelectric tensor

These are computed within ABINIT using DFPT methods. What are they? Conference on Computational Physics, Los Angeles, 2005 http://www.physics.rutgers.edu/~dhv/talks/ccp05.pdf

They are elements of “big Hessian matrix” Displacement Strain Displacement Strain E - field

E -field

K

-L

-Z

-L

C

-e

-Z

-e

c

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

Build from Elementary Tensors

To Relaxed-ion tensors

K mn

cab C jk Z ma L mj

c ab C jk eaj

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

Use relaxed ion cab , C jk , eaj to compute (h ) C (jkD ) = C (jke ) + eaj bab ebk

Elastic tensor at fixed D

(s ) cab = cab + eaj (C -1 ) jk ebk

Free-stress dielectric tensor

S jk = (C -1 ) jk

Elastic compliance tensor

b (h ) = (e (h ) )( -1)

Inverse dielectric tensor

daj = S (jke ) eak (s ) g aj = bab d bj

Various piezoelectric tensors

(h ) haj = bab ebj

kaj =

d aj

s

(s ) aa

S

(e ) jj

Electromechanical coupling

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

Elastic tensors at different elec. BC’s: ZnO C(e) (GPa) 226

139

123

0

0

0

139

226

123

0

0

0

123

123

242

0

0

0

0

0

0

40

0

0

0

0

0

0

40

0

0

0

0

0

0

44

Apply strain perturbation

Metallic

Short circuit boundary condition Metallic Measuring stress response and get

C(D) (GPa) 231

144

114

0

0

0

144

231

114

0

0

0

114

114

260

0

0

0

0

0

0

43

0

0

0

0

0

0

43

0

0

0

0

0

0

44

Metallic

C(e)

Apply strain perturbation Open circuit boundary condition

Metallic Measuring stress response and get

C(D)

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

Mapping Energy vs. Polarization

BaTiO3

Conference Dieguez on Computational Physics, Los Angeles, 2005 Oswaldo (W26.7 3:42pm Thursday) 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)

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