Berry Phases and Curvatures in Electronic-Structure Theory David - PowerPoint PPT Presentation

Berry Phases and Curvatures in Electronic-Structure Theory David Vanderbilt Rutgers University March APS Meeting, Baltimore, March 13 2006

Rahman prize for: – Theory of polarization (King-Smith & Vanderbilt) – Ultrasoft pseudopotentials Three quick preliminaries: • Who was Aneesur Rahman? • Who is Dominic King-Smith? • A parable about referee reports… March APS Meeting, Baltimore, March 13 2006

Who was Aneesur Rahman? “Father of Molecular Dynamics” • �Born Hyderbad, India • Educ. Cambridge, Louvain • Argonne Natl. Labs 1960-85 • U. Minnesota 1985-87 • Died 1987 • Rahman Prize established in 1992 with funds from IBM Photo courtesy Sam Bader via Marie-Louise Saboungi March APS Meeting, Baltimore, March 13 2006

Who is Dominic King-Smith? “Father of Bettina” • PhD, Cambridge, UK • Postdoc at Rutgers `91-`93 • Biosym/MSI/Accelrys `93-`01 • Presently at: Accelrys Job title: “Product Manager, Quantum Mechanics” March APS Meeting, Baltimore, March 13 2006

Ultrasoft Pseudopotentials March APS Meeting, Baltimore, March 13 2006

Berry Phases and Curvatures in Electronic-Structure Theory David Vanderbilt Rutgers University March APS Meeting, Baltimore, March 13 2006

Introduction • By mid-1990s, density-functional perturbation theory allowed calculation of linear response to E-field • However, it was not known how to: – Calculate polarization itself – Treat finite E-fields • Analogous problem of calculating orbital magnetization also unsolved March APS Meeting, Baltimore, March 13 2006

Introduction • Solutions of these problems are now in hand – Modern theory of polarization (1993) – Treatment of finite E-fields (2002) – Orbital magnetization (2005) • Solutions rely heavily on two crucial ingredients: – Wannier functions – Berry phases and related quantities This talk: Brief survey of methods! Almost nothing on applications March APS Meeting, Baltimore, March 13 2006

Outline of Talk • Introduction • Berry phases, potentials, and curvatures • Realizations: – Electric polarization – Wannier functions – Electric fields – Anomalous Hall conductivity – Orbital magnetization • Summary and prospects March APS Meeting, Baltimore, March 13 2006

Berry phases  u 4 Ò  u 3 Ò Now take limit  u 2 Ò that density of points Æ ∞  u n Ò =  u 1 Ò  u n-1 Ò … March APS Meeting, Baltimore, March 13 2006

Berry phases  u l Ò Continuum limit l =1 l =0 March APS Meeting, Baltimore, March 13 2006

(Context: Molecular coordinates) z 2  u l Ò Na 3 l =1 l =0 ( z 1 , z 2 ) z 1 March APS Meeting, Baltimore, March 13 2006

Context: k-space in Brillouin zone  u k Ò k y l =1 l =0 Bloch function k x 0 2 p /a March APS Meeting, Baltimore, March 13 2006

Stokes theorem: Berry curvature  u k Ò W k y k x 2 p /a 0 March APS Meeting, Baltimore, March 13 2006

Context: k-space in Brillouin zone  u k Ò k y l =1 l =0 Bloch function k x 0 2 p /a March APS Meeting, Baltimore, March 13 2006

Spanning the BZ l =0 l =1  u k Ò k y Bloch function k x 0 2 p /a March APS Meeting, Baltimore, March 13 2006

Does any of this have any connection to real physics of materials? March APS Meeting, Baltimore, March 13 2006

Outline of Talk • Introduction • Berry phases, potentials, and curvatures • Realizations: – Electric polarization – Wannier functions – Electric fields – Anomalous Hall conductivity – Orbital magnetization • Summary and prospects March APS Meeting, Baltimore, March 13 2006

P = d cell / V cell ? • Textbook picture (Claussius-Mossotti) + + + – – – • But does not correspond to reality! + + + – – – March APS Meeting, Baltimore, March 13 2006

Ferroelectric PbTiO 3 (Courtesy N. Marzari)

P = d cell / V cell ? d cell = March APS Meeting, Baltimore, March 13 2006

P = d cell / V cell ? d cell = March APS Meeting, Baltimore, March 13 2006

Berry-phase theory of electric polarization March APS Meeting, Baltimore, March 13 2006

Berry-phase theory of electric polarization Berry potential! March APS Meeting, Baltimore, March 13 2006

Simplify: 1 band, 1D l =0 l =1 k y  u k Ò k x 2 p /a 0 March APS Meeting, Baltimore, March 13 2006

Discrete sampling of k-space March APS Meeting, Baltimore, March 13 2006

Discretized formula in 3D where March APS Meeting, Baltimore, March 13 2006

Sample Application: Born Z * +2 e ? +4 e ? – 2 e ? – 2 e ? Paraelectric Ferroelectric March APS Meeting, Baltimore, March 13 2006

Outline of Talk • Introduction • Berry phases, potentials, and curvatures • Realizations: – Electric polarization – Wannier functions – Electric fields – Anomalous Hall conductivity – Orbital magnetization • Summary and prospects March APS Meeting, Baltimore, March 13 2006

Wannier function representation (Marzari and Vanderbilt, 1997) “Wannier center” March APS Meeting, Baltimore, March 13 2006

Mapping to Wannier centers Wannier center r n March APS Meeting, Baltimore, March 13 2006

Mapping to Wannier centers Wannier dipole theorem D P = S ion (Z ion e) D r ion + S wf (– 2e) D r wf • E xact! • Gives local description of dielectric response! March APS Meeting, Baltimore, March 13 2006

Ferroelectric BaTiO 3 (Courtesy N. Marzari)

Wannier functions Wannier functions in a-Si in l-H 2 O Fornari et al. Silvestrelli et al.

Wannier analysis of PVDF polymers and copolymers Courtesy S. Nakhmanson

Note upcoming release of public max-loc Wannier code… (Organized by Nicola Marzari) March APS Meeting, Baltimore, March 13 2006

Outline of Talk • Introduction • Berry phases, potentials, and curvatures • Realizations: – Electric polarization – Wannier functions – Electric fields – Anomalous Hall conductivity – Orbital magnetization • Summary and prospects March APS Meeting, Baltimore, March 13 2006

Electric Fields: The Problem Easy to do in practice: But ill-defined in principle: Zener tunneling For small E-fields, t Zener >> t Universe ; is it OK? March APS Meeting, Baltimore, March 13 2006

Electric Fields: The Problem y (x) is very messy • is not periodic • Bloch’s theorem does not apply March APS Meeting, Baltimore, March 13 2006

Electric Fields: The Solution • Seek long-lived resonance • Described by Bloch functions • Minimizing the electric enthalpy functional (Nunes and Gonze, 2001) Usual E KS Berry phase polarization Souza, Iniguez, and Vanderbilt, PRL 89, 117602 (2002); P. Umari and A. Pasquarello, PRL 89, 157602 (2002). March APS Meeting, Baltimore, March 13 2006

Electric Fields: Implementation As long as D k is not too small: • Can use standard methods to find minimum • The e · P term introduces coupling between k-points – p /a 0 p /a k March APS Meeting, Baltimore, March 13 2006

Sample Application: Born Z * Can check that previous results for BaTiO 3 are reproduced March APS Meeting, Baltimore, March 13 2006

Outline of Talk • Introduction • Berry phases, potentials, and curvatures • Realizations: – Electric polarization – Wannier functions – Electric fields – Anomalous Hall conductivity – Orbital magnetization • Summary and prospects March APS Meeting, Baltimore, March 13 2006

Anomalous Hall effect Ferromagnetic Material March APS Meeting, Baltimore, March 13 2006

Anomalous Hall effect • Karplus-Luttinger theory (1954) Semiclassical equations of motion: – Scattering-free, intrinsic • Skew-scattering mechanism (1955) – Impurity scattering • Side-jump mechanism (1970) – Impurity or phonon scattering • Berry-phase theory (1999) Sundaram and Niu, PRB 59, 14925 (1999). – Restatement of Karplus-Luttinger March APS Meeting, Baltimore, March 13 2006

Stokes theorem: Berry curvature  u k Ò W k y k x 2 p /a 0 March APS Meeting, Baltimore, March 13 2006

Anomalous Hall conductivity of SrRuO 3 W z for k z =0 Z. Fang et al, Science 302, 92 (2003). March APS Meeting, Baltimore, March 13 2006

X. Wang, J. Yates, I. Souza, and D. Vanderbilt, G23.00001 (Tuesday 8am). W z (k x ,k z ) in bcc Fe See also Y.G. Yao et al., PRL 92, 037204 (2004). March APS Meeting, Baltimore, March 13 2006

Outline of Talk • Introduction • Berry phases, potentials, and curvatures • Realizations: – Electric polarization – Wannier functions – Electric fields – Anomalous Hall conductivity – Orbital magnetization • Summary and prospects March APS Meeting, Baltimore, March 13 2006

Orbital Magnetization M is a bulk property? fl K = M x n fl K K is only apparently a surface property? P is a bulk property - s + s fl s = P ⋅ n fl s is only apparently a surface property March APS Meeting, Baltimore, March 13 2006