All electron DFT with FLEUR – hands-on tutorial 2017 · transport phenomena Anomalous Hall effect Jan-Philipp Hanke – [email protected] 17 kx ky Berry curvature @Hnm(k) @kj

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Wannier functions

Jan-Philipp Hanke

Peter Grünberg Institut and Institute for Advanced Simulation Forschungszentrum Jülich and JARA, 52425 Jülich, Germany

All electron DFT with FLEUR – hands-on tutorial 2017

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1.  What are Wannier functions? v  definition and basic properties v  gauge and localization

2.  How to construct them starting from DFT? v  Marzari-Vanderbilt algorithm v  FLEUR interface to wannier90

3.  Where to apply them? v  electronic band structure v  anomalous Hall effect v  ferroelectric polarization

Outline of the lecture

Jan-Philipp Hanke – [email protected] 2

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v  electronic-structure problem

What are Wannier functions?


kn(r) = eik·rukn(r)

H | kni = Ekn | kni

|WRni =1

N

X

k

e�ik·RX

m

U (k)mn| kni

v  alternative localized solution?

v  Fourier transformation of some group of Bloch states

delocalized Bloch waves

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4 Jan-Philipp Hanke – [email protected]

Rev. Mod. Phys 84, 1419 (2012)

Wannier functions for silicon

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What are they useful for?


minimal models

molecular dynamics

mean-field theory

anomalous Hall effect

spin Hall effect

ferroelectric polarization

orbital magnetism

spin-orbit torques

Dzyaloshinskii-Moriya interaction

electron-phonon coupling

chemical bondings

disordered systems

hybridization chemistry

… and many more

Wannier functions

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v  ideal to visualize chemical bonding

v  are not eigenstates of Hamiltonian

v  direct lattice vector as label

What are their properties?


WRn(r) = W0n(r �R)

R

�(r) =1

(2⇡)3

Zeik·rd3k

v  localized in real space (similar to wave packets)

are WFs always localized and unique?

bonding anti-bonding

Rev. Mod. Phys 84, 1419 (2012)

W0(z)

WR1(z)

WR2(z)

W0n

WR1n

WR2n

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Gauge freedom of Bloch states


| kni ! ei�n(k)| kni

| kni !X

m

U (k)mn| kmi

v  WFs are not unique!

v  more localized if Bloch states evolve smoothly with

gauge A

Freimuth, PhD thesis, FZJ

gauge B

k

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Generating maximally-localized Wannier functions (MLWFs)

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v  eliminate gauge freedom by exploiting mixing of bands

“Make WFs unique again”


|WRni =1

N

X

k

e�ik·RX

m

U (k)mn| kni

v  different criteria to find optimally-smooth gauge

functional of gauge U (k)

mn

hrin = hW0n|r|W0ni

⌦ =X

n

⌦(r � hrin)2

↵n=

X

n

(hr2in � hri2n)

Maximal localization by minimizing spatial spread ⌦

hr2in = hW0n|r2|W0ni

Alternative: maximize on-site Coulomb matrix elements

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Marzari-Vanderbilt iterative algorithm


hWRn|r|W0mi = iV

(2⇡)3

Zd3k eik·Rhukn|rk|ukmi

gauge enters here

U (k)mn

|W̃0ni =1

N

X

km

| kmih km|gni

trial orbitals such as s, p,… or hybrids

initial guess for gauge

determine new gauge

compute spread and gradient . @⌦/@U (k)

mn

finite differences

kk � b k + b

wbwb

rkf(k) =X

bwbb [f(k + b)� f(k)]

hW0n|r|W0ni = � 1

N

X

kb

wbb Im lnM (k,b)nn

requires overlaps at neighboring k-points M (k,b)

mn = hukm|uk+bni

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Compute MLWFs based on following input from a DFT code

The wannier90 code


Some features of wannier90: v  Generation and plotting of MLWFs v  Hopping parameters

v  Band structure, anomalous Hall effect, Berry curvature

v  Interpolation of orbital magnetization

http://www.wannier.org

A(k)mn = h km|gni

M (k,b)mn = hukm|uk+bni

(starting guess)

EknBloch eigenvalues

(centers and spreads)

further information (Bravais matrix, k-points,…)

WF1.amn:

WF1.mmn:

WF1.eig:

WF1.win:

Hnm(R1 �R2) = hWR1n|H|WR2mi

decay rapidly with distance

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The FLEUR interface to wannier90


self-consistent calculation of charge density in

FLEUR

prepare Wannier step

à WF1.win, bkpts, proj

generate necessary matrices for MLWFs

à WF1.amn, WF1.mmn,

WF1.eig

Wannierization using wannier90 code

à WF1.wout, WF1.chk

post-processing (FLEUR/wannier90)

http://www.wannier.org

GaAs Si PRB 78, 035120 (2008)

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Selected applications

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v  repetitions due to discrete k-mesh

v  interpolation scheme will rely on fast convergence of MLWFs with number of k-points

Wannier interpolation


0

0.12

0.24

0.36

�8 �6 �4 �2 0 2 4 6 8

|W(z)|2

position z along the chain direction in units of a

Nk = 8

Nk ! 1

z

=1

Nk

1� e2⇡iz

1� e2⇡iz/Nk

Nk!1�! �1

2e⇡izj0(⇡z)

Example: plane waves in 1D

Bessel function

W (z) =1

Nk

Xkeikz

supercell

Nk = 8

Nk ! 1

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2⇡/a L

Fk

2⇡/a

F�1R

k

H(k)

R

H(R)

q

H(q)

H(k) H(R) H(q)

FT FT

8⇥ 8⇥ 8 200⇥ 200⇥ 200

Wannier interpolation


|'kni =X

m

U (k)nm| kmi

Aguilera & Friedrich, 48th IFF Spring School 2017 on topological matter

convergence of shape of MLWFs

Hnm(q) = h'qn|H|'qmiws =1

N

X

RR0

eiq·(R0�R)hWRn|H|WR0miws

“whole space” all

=X

R

eiq·RhW0n|H|WRmiws ⇡X

R

eiq·RhW0n|H|WRmisc

“supercell”

all

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v  bands overlap and hybridize with others

Metallic energy bands


v  construct set of MLWFs from larger group of bands, e.g., 18 MLWFs out of 28 energy bands

Palladium

frozen window

sp3d2

dxy

Freimuth, Wannier tutorial 2014

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V

VHm

SOC

MLWFs enable us to study non-trivial transport phenomena

Anomalous Hall effect


kx

ky

Berry curvature

@Hnm(k)

@kj= �i

X

R

Rj e�ik·RHnm(R)

derivatives of the Hamiltonian are easy to evaluate from WFs:

⌦xy

(k) = �2~2 ImoccX

n

X

m 6=n

hukm|vx

|uknihukn|vy|ukmi(Ekm � Ekn)2

Kubo-like formula:

anomalous Hall effect

⌦xy

(k) = 2ImX

n

⌧@ukn@k

x

��@ukn@k

y

�

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Anomalous Hall effect


bcc Fe

theory: experiment: �xy

= 757 (⌦cm)�1 �xy

⇡ 1000 (⌦cm)�1

�xy

= �e2

~

Zd3k

(2⇡)3⌦xy

(k)

ultra-dense k-mesh required

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Freimuth et al., PRB 78, 035120 (2008)

O

Ti

Ba

Ferroelectric polarization


Resta, Ferroelectrics 136, 51 (1992) King-Smith, Vanderbilt, PRB 47, 1651 (1993)

for finite systems only! P =d

V= � |e|

V

Zr n(r) d3r

v  operator unbounded in crystal

v  only polarization changes are physically relevant/observable

r

Pel

= � |e|V

occX

n

hW0n|r|W0ni

v  Berry phase theory: polarization from centers of MLWFs

w.r.t. centrosymmetric reference

P

P

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Step-by-step guide


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1.  What are Wannier functions? v  definition and basic properties v  gauge and localization

2.  How to construct them starting from DFT? v  Marzari-Vanderbilt algorithm v  FLEUR interface to wannier90

3.  Where to apply them? v  electronic band structure v  anomalous Hall effect v  ferroelectric polarization

Summary


Review: Rev. Mod. Phys 84, 1419 (2012)

Further reading: PRB 74, 195118 (2006) PRB 75, 195121 (2007) PRB 76, 195109 (2007) PRB 78, 035120 (2008)

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Thank you for your attention!

Documents

All electron DFT with FLEUR – hands-on tutorial 2017 · transport phenomena Anomalous Hall effect Jan-Philipp Hanke – [email protected] 17 kx ky Berry curvature @Hnm(k) @kj