Theory of Polarization: A Modern Approach

  • Raffaele RestaEmail author
  • David Vanderbilt
Part of the Topics in Applied Physics book series (TAP, volume 105)


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.


Brillouin Zone Lattice Vector Spontaneous Polarization Modern Theory Modern Approach 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. L. D. Landau, E. M. Lifshitz: Electrodynamics of Continuous Media (Pergamon, Oxford 1984) Google Scholar
  2. 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) Google Scholar
  3. R. Clausius: Die Mechanische Behandlung der Electrica (Vieweg, Berlin 1879) Google Scholar
  4. 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) CrossRefGoogle Scholar
  5. S. Lundqvist, N. H. March (Eds.): Theory of the Inhomogeneous Electron Gas (Plenum, New York 1983) Google Scholar
  6. W. E. Pickett: Pseudopotential methods in condensed matter applications, Comput. Phys. Rep. 9, 115 (1989) CrossRefGoogle Scholar
  7. C. Kittel: Introduction to Solid State Physics, 7 ed. (Wiley, New York 1996) Google Scholar
  8. N. W. Ashcroft, N. D. Mermin: Solid State Physics (Saunders, Philadelphia 1976) Google Scholar
  9. R. M. Martin: Comment on calculations of electric polarization in crystals, Phys. Rev. B 9, 1998 (1974) CrossRefGoogle Scholar
  10. R. M. Martin: Piezoelectricity, Phys. Rev. B 5, 1607 (1972) CrossRefGoogle Scholar
  11. R. M. Martin: Comment on piezoelectricity under hydrostatic pressure, Phys. Rev. B 6, 4874 (1972) CrossRefGoogle Scholar
  12. W. F. Woo, W. Landauer: Comment on ``piezoelectricity under hydrostatic pressure'', Phys. Rev. B 6, 4876 (1972) CrossRefGoogle Scholar
  13. R. Landauer: Pyroelectricity and piezoelectricity are not true volume effects, Solid State Commun. 40, 971 (1981) CrossRefGoogle Scholar
  14. C. Kallin, B. J. Halperin: Surface-induced charge disturbances and piezoelectricity in insulating crystals, Phys. Rev. B 29, 2175 (1984) CrossRefGoogle Scholar
  15. R. Landauer: Introduction to ferroelectric surfaces, Ferroelectrics 73, 41 (1987) CrossRefGoogle Scholar
  16. A. K. Tagantsev: Electric polarization in crystals and its response to thermal and elastic perturbations, Phase Transitions 35, 119 (1991) Google Scholar
  17. R. Resta: Theory of the electric polarization in crystals, Ferroelectrics 136, 51 (1992) CrossRefGoogle Scholar
  18. R. D. King-Smith, D. Vanderbilt: Theory of polarization of crystalline solids, Phys. Rev. B 47, 1651 (1993) CrossRefGoogle Scholar
  19. R. Resta: Macroscopic polarization in crystalline dielectrics: The geometric phase approach, Rev. Mod. Phys. 66, 899 (1994) CrossRefGoogle Scholar
  20. 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) CrossRefGoogle Scholar
  21. A. Shapere, F. Wilczek (Eds.): Geometric Phases in Physics (World Scientific, Singapore 1989) Google Scholar
  22. R. Resta: Manifestations of {B}erry's phase in molecules and condensed matter, J. Phys. Condens. Matter 12, R107 (2000) CrossRefGoogle Scholar
  23. D. Vanderbilt, R. D. King-Smith: Electric polarization as a bulk quantity and its relation to surface charge, Phys. Rev. B 48, 4442 (1993) CrossRefGoogle Scholar
  24. D. J. Thouless: Quantization of particle transport, Phys. Rev. B 27, 6083 (1983) CrossRefGoogle Scholar
  25. R. Resta, M. Posternak, A. Baldereschi: Towards a quantum theory of polarization in ferroelectrics: The case of KNbO3, Phys. Rev. Lett. 70, 1010 (1993) CrossRefGoogle Scholar
  26. 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) CrossRefGoogle Scholar
  27. P. Ghosez, J.-P. Michenaud, X. Gonze: Dynamical atomic charges: The case of ABO3 compounds, Phys. Rev. B 58, 6224 (1998) CrossRefGoogle Scholar
  28. R. Pick, M. H. Cohen, R. M. Martin: Microscopic theory of force constants in the adiabatic approximation, Phys. Rev. B 1, 910 (1970) CrossRefGoogle Scholar
  29. M. Born, K. Huang: Dynamical Theory of Crystal Lattices (Oxford University Press, Oxford 1954) Google Scholar
  30. J. D. Axe: Apparent ionic charges and vibrational eigenmodes of BaTiO3 and other perovskites, Phys. Rev. 157, 429 (1967) CrossRefGoogle Scholar
  31. R. Resta: Dynamical charges in oxides: {R}ecent advances, J. Phys. Chem. Solids 61, 153 (1999) CrossRefGoogle Scholar
  32. W. Zhong, R. D. {King-Smith}, D. Vanderbilt: Giant {L}{O}-{T}{O} splittings in perovskite ferroelectrics, Phys. Rev. Lett. 72, 3618 (1994) CrossRefGoogle Scholar
  33. 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) CrossRefGoogle Scholar
  34. 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) CrossRefGoogle Scholar
  35. O. H. Nielsen, R. M. Martin: First-principles calculation of stress, Phys. Rev. Lett. 50, 697 (1983) CrossRefGoogle Scholar
  36. O. H. Nielsen, R. M. Martin: Quantum-mechanical theory of stress and force, Phys. Rev. B 32, 3780 (1985) CrossRefGoogle Scholar
  37. 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) CrossRefGoogle Scholar
  38. G. S\'aghi-Szab\'o, R. E. Cohen, H. Krakauer: First-principles study of piezoelectricity in PbTiO3, Phys. Rev. Lett. 80, 4321 (1998) CrossRefGoogle Scholar
  39. H. Fu, R. E. Cohen: Polarization rotation mechanism for ultrahigh electromechanical response in single-crystal piezoelectrics, Nature 403, 281 (2000) CrossRefGoogle Scholar
  40. 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) CrossRefGoogle Scholar
  41. 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) CrossRefGoogle Scholar
  42. D. Vanderbilt: Berry-phase theory of proper piezoelectric response, J. Phys. Chem. Solids 61, 147 (2000) CrossRefGoogle Scholar
  43. R. W. Nunes, D. Vanderbilt: Real-space approach to calculation of electric polarization and dielectric constants, Phys. Rev. Lett. 73, 712 (1994) CrossRefGoogle Scholar
  44. R. W. Nunes, X. Gonze: Berry-phase treatment of the homogeneous electric field perturbation in insulators, Phys. Rev. B 63, 155107 (2001) CrossRefGoogle Scholar
  45. I. Souza, J. \`I{\~n}iguez, D. Vanderbilt: First-principles approach to insulators in finite electric fields, Phys. Rev. Lett. 89, 117602 (2002) CrossRefGoogle Scholar
  46. P. Umari, A. Pasquarello: Ab initio molecular dynamics in a finite homogeneous electric field, Phys. Rev. Lett. 89, 157602 (2002) CrossRefGoogle Scholar
  47. 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) CrossRefGoogle Scholar
  48. 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) CrossRefGoogle Scholar
  49. G. Ortiz, R. M. Martin: Macroscopic polarization as a geometric quantum phase: Many-body formulation, Phys. Rev. B 49, 14 202 (1994) CrossRefGoogle Scholar
  50. R. Resta: Quantum-mechanical position operator in extended systems, Phys. Rev. Lett. 80, 1800 (1998) CrossRefGoogle Scholar
  51. I. Souza, T. Wilkens, R. M. Martin: Polarization and localization in insulators: Generating function approach, Phys. Rev. B 62, 1666 (2000) CrossRefGoogle Scholar
  52. R. Resta: Why are insulators insulating and metals conducting?, J. Phys. Condens. Matter 14, R625 (2002) CrossRefGoogle Scholar
  53. 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) CrossRefGoogle Scholar
  54. X. Gonze, P. Ghosez, R. W. Godby: Density-functional theory of polar insulators, Phys. Rev. Lett. 78, 294 (1997) CrossRefGoogle Scholar
  55. D. Vanderbilt: Nonlocality of {K}ohn-{S}ham exchange-correlation fields in dielectrics, Phys. Rev. Lett. 79, 3966 (1997) CrossRefGoogle Scholar
  56. W. Kohn: Theory of the insulating state, Phys. Rev. 133, A171 (1964) CrossRefGoogle Scholar
  57. N. Marzari, D. Vanderbilt: Maximally localized generalized {W}annier functions for composite energy bands, Phys. Rev. B 56, 12847 (1997) CrossRefGoogle Scholar
  58. R. Resta, S. Sorella: Electron localization in the insulating state, Phys. Rev. Lett. 82, 370 (1999) CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2007

Authors and Affiliations

  1. 1.INFM–DEMOCRITOS National Simulation CenterTriesteItaly
  2. 2.Dipartimento di Fisica TeoricaUniversità di TriesteTriesteItaly
  3. 3.Department of Physics and AstronomyRutgers UniversityPiscatawayUSA

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