ICTP/Psi-k/CECAM School on Electron-Phonon Physics from First - PowerPoint PPT Presentation

ICTP/Psi-k/CECAM School on Electron-Phonon Physics from First Principles Trieste, 19-23 March 2018

Lecture Tue.2 Maximally-localized Wannier functions Giovanni Pizzi 1 , Antimo Marrazzo 1 , Valerio Vitale 2 1 Ti eory and Simulation of Materials, EPFL (Switzerland) 2 Cavendish Laboratory, Department of Physics, University of Cambridge (UK) School on Electron-Phonon Physics from First Principles Trieste, March 20th, 2018

References • Marzari, N., and D. Vanderbilt, Phys. Rev. B 56, 12847 (1997) • Souza, I., N. Marzari, and D. Vanderbilt, Phys. Rev. B 65, 035109 (2001) • N. Marzari et al., Rev. Mod. Phys. 84, 1419–1475 (2012) • R. M. Martin, Electronic Structure: Basic Theory and Practical Methods, Cambridge, 2004 • www.wannier.org • First part of the slides: courtesy of Prof. Nicola Marzari. 
 Can be found on the Wannier90 website: www.wannier.org under 
 User Guide > NSF Summer School 2009 > N. Marzari Lecture Slides

PART I Wannier functions

Bloch Theorem Crystal in real space: Brillouin zone in reciprocal space: – π /a 0 π /a k Courtesy of I. Souza / D. Vanderbilt

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 unit cell and periodic 
 Brillouin zone in reciprocal space: over the cells If there is only one band ( n =1): Ψ k ( r ) = u k ( r ) e i k · r And we can define the Wannier functions: Z Ψ k ( r ) e − i k · R d k | R i = BZ – π /a 0 π /a k - One WF per lattice vector R: N in total with Born-von Karman PBC 
 Courtesy of I. Souza / D. Vanderbilt with N total unit cells - They are all identical, only shifted: if we have they are 
 | R 1 i , | R 2 i shifted by R 2 - R 1

From Bloch Orbitals to Wannier Func:ons Multiband case, simplest thing to do: Note : The shape of the 
 WFs (in real space) will be different for every phase!

From Bloch Orbitals to Wannier Func:ons Multiband case, simplest thing to do: More generally:

Orthogonal and unitary transforma:ons Unitary matrix Rotated Bloch function n=2 n=1 –  /a 0  /a – π /a 0 π /a k k Courtesy of I. Souza / D. Vanderbilt

Generalized Wannier Func:ons for Composite Bands Each unitary matrix chooses a different set of WFs. We would like to choose the “best”, i.e. the “maximally-localized”

The Localiza:on Func:onal (Foster‐Boys) N. Marzari and D. Vanderbilt, Phys. Rev. B 56, 12847 (1997)

Decomposi:on of the Localiza:on Func:onal

How to compute? Blount identities Centers of Wannier func:ons: definition Bloch theorem WF center

Blount iden::es Therefore: Numerical approach: numerical derivatives on a uniform k grid in the BZ We can express the 
 relevant quantities as 
 a function of the M mn matrices (these will be one of the main 
 inputs to Wannier90)

To compute the maximal localization, 
 we do not need to know the wavefunctions , but only the overlaps M mn matrices at neighbouring k-points (after minimization, if we want to plot the Wannier functions in real space, we need instead to know the u nk - in the code: files UNK) Numerical approach: numerical derivatives on a uniform k grid in the BZ We can express the 
 relevant quantities as 
 a function of the M mn matrices (these will be one of the main 
 inputs to Wannier90)

Silicon, GaAs, Amorphous Silicon, Benzene M. Fornari, N. Marzari, M. Peressi, and A. Baldereschi, Comp. Mater. Science 20, 337 (2001)

The localisation procedure • Long-range decay : Wannier functions corresponding 
 to isolated valence bands decay to zero 
 exponentially with the distance from their center • At the global minimum (maximally-localized WFs) 
 the Wannier functions are real 
 (the code prints the max. absolute ratio of 
 imaginary and real part to check this) • We might find a local minimum! Care is needed 
 • If we expect (from physical/chemical considerations) 
 the shape and position of Wannier functions, we can 
 give an initial guess in the form of projections on 
 localised orbitals

Real‐Space Projectors