Polarization Integration
Transcript of 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
Conference on Computational Physics, Los Angeles, 2005http://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, 2005http://www.physics.rutgers.edu/~dhv/talks/ccp05.pdf
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
Conference on Computational Physics, Los Angeles, 2005http://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).
Conference on Computational Physics, Los Angeles, 2005http://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, 2005http://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 onapplications
Conference on Computational Physics, Los Angeles, 2005http://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
Conference on Computational Physics, Los Angeles, 2005http://www.physics.rutgers.edu/~dhv/talks/ccp05.pdf
P = dsample / Vsample ?
+s-s
DP = ( L2 s ) . L / L3
L x L x L sample:
Conference on Computational Physics, Los Angeles, 2005http://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
Conference on Computational Physics, Los Angeles, 2005http://www.physics.rutgers.edu/~dhv/talks/ccp05.pdf
P = dcell / Vcell ?
+–
+–
+–
+–
+–
+–
• Textbook picture(Claussius-Mossotti)
• But does not correspondto reality!
Ferroelectric PbTiO3 (Courtesy N. Marzari)
Conference on Computational Physics, Los Angeles, 2005http://www.physics.rutgers.edu/~dhv/talks/ccp05.pdf
P = dcell / Vcell ?
dcell = 0
Conference on Computational Physics, Los Angeles, 2005http://www.physics.rutgers.edu/~dhv/talks/ccp05.pdf
P = dcell / Vcell ?
dcell =
Conference on Computational Physics, Los Angeles, 2005http://www.physics.rutgers.edu/~dhv/talks/ccp05.pdf
P = dcell / Vcell ?
dcell =
Conference on Computational Physics, Los Angeles, 2005http://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
Conference on Computational Physics, Los Angeles, 2005http://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
Conference on Computational Physics, Los Angeles, 2005http://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
Conference on Computational Physics, Los Angeles, 2005http://www.physics.rutgers.edu/~dhv/talks/ccp05.pdf
Attempt 4
Conference on Computational Physics, Los Angeles, 2005http://www.physics.rutgers.edu/~dhv/talks/ccp05.pdf
Conference on Computational Physics, Los Angeles, 2005http://www.physics.rutgers.edu/~dhv/talks/ccp05.pdf
Simplify: 1 band, 1D
Conference on Computational Physics, Los Angeles, 2005http://www.physics.rutgers.edu/~dhv/talks/ccp05.pdf
Discrete sampling of k-space
Conference on Computational Physics, Los Angeles, 2005http://www.physics.rutgers.edu/~dhv/talks/ccp05.pdf
Gauge invariance
Conference on Computational Physics, Los Angeles, 2005http://www.physics.rutgers.edu/~dhv/talks/ccp05.pdf
Discretized formula in 3D
where
Conference on Computational Physics, Los Angeles, 2005http://www.physics.rutgers.edu/~dhv/talks/ccp05.pdf
Sample Application: Born Z*
Paraelectric Ferroelectric
+2 e ?
+4 e ?
– 2 e ?
– 2 e ?
Conference on Computational Physics, Los Angeles, 2005http://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, 2005http://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
Conference on Computational Physics, Los Angeles, 2005http://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
on eigenfunctions• not bounded from below• There is no ground state
y(x) is verymessy
Conference on Computational Physics, Los Angeles, 2005http://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, 2005http://www.physics.rutgers.edu/~dhv/talks/ccp05.pdf
Electric Fields: Justification
Seeklong-livedmetastable
periodicsolution
Conference on Computational Physics, Los Angeles, 2005http://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
Conference on Computational Physics, Los Angeles, 2005http://www.physics.rutgers.edu/~dhv/talks/ccp05.pdf
Sample Application: Born Z*
Can check that previous resultsfor BaTiO3 are reproduced
Conference on Computational Physics, Los Angeles, 2005http://www.physics.rutgers.edu/~dhv/talks/ccp05.pdf
Sample Application: Born Z*
(Souza,Iniguez,and Vanderbilt,
2002)
Conference on Computational Physics, Los Angeles, 2005http://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, 2005http://www.physics.rutgers.edu/~dhv/talks/ccp05.pdf
Wannier function representation
(Marzari andVanderbilt, 1997)
“Wannier center”
Conference on Computational Physics, Los Angeles, 2005http://www.physics.rutgers.edu/~dhv/talks/ccp05.pdf
Mapping to Wannier centers
Wanniercenter
rn
Conference on Computational Physics, Los Angeles, 2005http://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
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
Conference on Computational Physics, Los Angeles, 2005http://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, 2005http://www.physics.rutgers.edu/~dhv/talks/ccp05.pdf
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)
Conference on Computational Physics, Los Angeles, 2005http://www.physics.rutgers.edu/~dhv/talks/ccp05.pdf
- fieldEStrainDisplacement
-L
-L
c
-e-Z
-e
-ZK
C
- fieldE
Strain
Displacement
They are elements of “big Hessian matrix”
Conference on Computational Physics, Los Angeles, 2005http://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
abc
L j
jk
e
C
a
abc
Conference on Computational Physics, Los Angeles, 2005http://www.physics.rutgers.edu/~dhv/talks/ccp05.pdf
)()(
)(
)(
)(
)()()(
)(
)()()(
)(
)(
)(
,,
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
Conference on Computational Physics, Los Angeles, 2005http://www.physics.rutgers.edu/~dhv/talks/ccp05.pdf
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
Conference on Computational Physics, Los Angeles, 2005http://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, 2005http://www.physics.rutgers.edu/~dhv/talks/ccp05.pdf
Mapping Energy vs. Polarization
BaTiO3 (Courtesy N. Marzari)
Oswaldo Dieguez (W26.7 3:42pm Thursday)
Conference on Computational Physics, Los Angeles, 2005http://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, 2005http://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