Abstract
In this Chapter we review the physical basis of the modern theory of polarization, emphasizing how the polarization can be defined in terms of the accumulated adiabatic flow of current occurring as a crystal is modified or deformed. We explain how the polarization is closely related to a Berry phase of the Bloch wavefunctions as the wavevector is carried across the Brillouin zone, or equivalently, to the centers of charge of Wannier functions constructed from the Bloch wavefunctions. A resulting feature of this formulation is that the polarization is formally defined only modulo a “quantum of polarization” – in other words, that the polarization may be regarded as a multi-valued quantity. We discuss the consequences of this theory for the physical understanding of ferroelectric materials, including polarization reversal, piezoelectric effects, and the appearance of polarization charges at surfaces and interfaces. In so doing, we give a few examples of realistic calculations of polarization-related quantities in perovskite ferroelectrics, illustrating how the present approach provides a robust and powerful foundation for modern computational studies of dielectric and ferroelectric materials.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
L. D. Landau, E. M. Lifshitz: Electrodynamics of Continuous Media (Pergamon, Oxford 1984)
O. F. Mossotti: Azioni e deformazioni nei dielettrici, Memorie di Matematica e di Fisica della Societ\`a Italiana delle Scienze Residente in Modena 24, 49 (1850)
R. Clausius: Die Mechanische Behandlung der Electrica (Vieweg, Berlin 1879)
M. Posternak, R. Resta, A. Baldereschi: Role of covalent bonding in the polarization of perovskite oxides: The case of {KNbO_3}, Phys. Rev. B 50, 8911 (1994)
S. Lundqvist, N. H. March (Eds.): Theory of the Inhomogeneous Electron Gas (Plenum, New York 1983)
W. E. Pickett: Pseudopotential methods in condensed matter applications, Comput. Phys. Rep. 9, 115 (1989)
C. Kittel: Introduction to Solid State Physics, 7 ed. (Wiley, New York 1996)
N. W. Ashcroft, N. D. Mermin: Solid State Physics (Saunders, Philadelphia 1976)
R. M. Martin: Comment on calculations of electric polarization in crystals, Phys. Rev. B 9, 1998 (1974)
R. M. Martin: Piezoelectricity, Phys. Rev. B 5, 1607 (1972)
R. M. Martin: Comment on piezoelectricity under hydrostatic pressure, Phys. Rev. B 6, 4874 (1972)
W. F. Woo, W. Landauer: Comment on ``piezoelectricity under hydrostatic pressure'', Phys. Rev. B 6, 4876 (1972)
R. Landauer: Pyroelectricity and piezoelectricity are not true volume effects, Solid State Commun. 40, 971 (1981)
C. Kallin, B. J. Halperin: Surface-induced charge disturbances and piezoelectricity in insulating crystals, Phys. Rev. B 29, 2175 (1984)
R. Landauer: Introduction to ferroelectric surfaces, Ferroelectrics 73, 41 (1987)
A. K. Tagantsev: Electric polarization in crystals and its response to thermal and elastic perturbations, Phase Transitions 35, 119 (1991)
R. Resta: Theory of the electric polarization in crystals, Ferroelectrics 136, 51 (1992)
R. D. King-Smith, D. Vanderbilt: Theory of polarization of crystalline solids, Phys. Rev. B 47, 1651 (1993)
R. Resta: Macroscopic polarization in crystalline dielectrics: The geometric phase approach, Rev. Mod. Phys. 66, 899 (1994)
S. Baroni, S. de Gironcoli, A. {Dal Corso}, P. Giannozzi: Phonons and related crystal properties from density-functional perturbation theory, Rev. Mod. Phys. 73, 515 (2001)
A. Shapere, F. Wilczek (Eds.): Geometric Phases in Physics (World Scientific, Singapore 1989)
R. Resta: Manifestations of {B}erry's phase in molecules and condensed matter, J. Phys. Condens. Matter 12, R107 (2000)
D. Vanderbilt, R. D. King-Smith: Electric polarization as a bulk quantity and its relation to surface charge, Phys. Rev. B 48, 4442 (1993)
D. J. Thouless: Quantization of particle transport, Phys. Rev. B 27, 6083 (1983)
R. Resta, M. Posternak, A. Baldereschi: Towards a quantum theory of polarization in ferroelectrics: The case of KNbO3, Phys. Rev. Lett. 70, 1010 (1993)
S. Dall'Olio, R. Dovesi, R. Resta: Spontaneous polarization as a {B}erry phase of the {H}artree-{F}ock wave function: The case of KNbO3, Phys. Rev. B 56, 10105 (1997)
P. Ghosez, J.-P. Michenaud, X. Gonze: Dynamical atomic charges: The case of ABO3 compounds, Phys. Rev. B 58, 6224 (1998)
R. Pick, M. H. Cohen, R. M. Martin: Microscopic theory of force constants in the adiabatic approximation, Phys. Rev. B 1, 910 (1970)
M. Born, K. Huang: Dynamical Theory of Crystal Lattices (Oxford University Press, Oxford 1954)
J. D. Axe: Apparent ionic charges and vibrational eigenmodes of BaTiO3 and other perovskites, Phys. Rev. 157, 429 (1967)
R. Resta: Dynamical charges in oxides: {R}ecent advances, J. Phys. Chem. Solids 61, 153 (1999)
W. Zhong, R. D. {King-Smith}, D. Vanderbilt: Giant {L}{O}-{T}{O} splittings in perovskite ferroelectrics, Phys. Rev. Lett. 72, 3618 (1994)
P. Ghosez, X. Gonze, P. Lambin, J.-P. Michenaud: Born effective charges of barium titanate: Band-by-band decomposition and sensitivity to structural features, Phys. Rev. B 51, 6765 (1995)
S. de Gironcoli, S. Baroni, R. Resta: Piezoelectric properties of {I}{I}{I}-{V} semiconductors from first-principles linear-response theory, Phys. Rev. Lett. 62, 2853 (1989)
O. H. Nielsen, R. M. Martin: First-principles calculation of stress, Phys. Rev. Lett. 50, 697 (1983)
O. H. Nielsen, R. M. Martin: Quantum-mechanical theory of stress and force, Phys. Rev. B 32, 3780 (1985)
O. H. Nielsen, R. M. Martin: Stresses in semiconductors: Ab initio calculations on {S}i, {G}e, and {G}a{A}s, Phys. Rev. B 32, 3792 (1985)
G. S\'aghi-Szab\'o, R. E. Cohen, H. Krakauer: First-principles study of piezoelectricity in PbTiO3, Phys. Rev. Lett. 80, 4321 (1998)
H. Fu, R. E. Cohen: Polarization rotation mechanism for ultrahigh electromechanical response in single-crystal piezoelectrics, Nature 403, 281 (2000)
G. S\'aghi-Szab\'o, R. E. Cohen, H. Krakauer: First-principles study of piezoelectricity in tetragonal PbTiO3 and PbZr1/2Ti1/2O3, Phys. Rev. B 59, 12771 (1999)
L. Bellaiche, D. Vanderbilt: Intrinsic piezoelectric response in perovskite alloys: {P}{M}{N}-{P}{T} versus {P}{Z}{T}, Phys. Rev. Lett. 83, 1347 (1999)
D. Vanderbilt: Berry-phase theory of proper piezoelectric response, J. Phys. Chem. Solids 61, 147 (2000)
R. W. Nunes, D. Vanderbilt: Real-space approach to calculation of electric polarization and dielectric constants, Phys. Rev. Lett. 73, 712 (1994)
R. W. Nunes, X. Gonze: Berry-phase treatment of the homogeneous electric field perturbation in insulators, Phys. Rev. B 63, 155107 (2001)
I. Souza, J. \`I{\~n}iguez, D. Vanderbilt: First-principles approach to insulators in finite electric fields, Phys. Rev. Lett. 89, 117602 (2002)
P. Umari, A. Pasquarello: Ab initio molecular dynamics in a finite homogeneous electric field, Phys. Rev. Lett. 89, 157602 (2002)
I. Souza, J. \`I{\~n}iguez, D. Vanderbilt: Dynamics of {B}erry-phase polarization in time-dependent electric fields, Phys. Rev. B 69, 085106 (2004)
J. \'I{\~n}iguez, L. Bellaiche, D. Vanderbilt: First-principles study of (BiScO3)1-x–(PbTiO3)x piezoelectric alloys, Phys. Rev. B 67, 224107 (2003)
G. Ortiz, R. M. Martin: Macroscopic polarization as a geometric quantum phase: Many-body formulation, Phys. Rev. B 49, 14 202 (1994)
R. Resta: Quantum-mechanical position operator in extended systems, Phys. Rev. Lett. 80, 1800 (1998)
I. Souza, T. Wilkens, R. M. Martin: Polarization and localization in insulators: Generating function approach, Phys. Rev. B 62, 1666 (2000)
R. Resta: Why are insulators insulating and metals conducting?, J. Phys. Condens. Matter 14, R625 (2002)
X. Gonze, P. Ghosez, R. W. Godby: Density-polarization functional theory of the response of a periodic insulating solid to an electric field, Phys. Rev. Lett. 74, 4035 (1995)
X. Gonze, P. Ghosez, R. W. Godby: Density-functional theory of polar insulators, Phys. Rev. Lett. 78, 294 (1997)
D. Vanderbilt: Nonlocality of {K}ohn-{S}ham exchange-correlation fields in dielectrics, Phys. Rev. Lett. 79, 3966 (1997)
W. Kohn: Theory of the insulating state, Phys. Rev. 133, A171 (1964)
N. Marzari, D. Vanderbilt: Maximally localized generalized {W}annier functions for composite energy bands, Phys. Rev. B 56, 12847 (1997)
R. Resta, S. Sorella: Electron localization in the insulating state, Phys. Rev. Lett. 82, 370 (1999)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Copyright information
© 2007 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Resta, R., Vanderbilt, D. (2007). Theory of Polarization: A Modern Approach. In: Physics of Ferroelectrics. Topics in Applied Physics, vol 105. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-34591-6_2
Download citation
DOI: https://doi.org/10.1007/978-3-540-34591-6_2
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-34590-9
Online ISBN: 978-3-540-34591-6
eBook Packages: Physics and AstronomyPhysics and Astronomy (R0)