Wannier Functions: ab-initio tight-binding

Jonathan Yates

Cavendish Laboratory, Cambridge University

Representing a Fermi Surface

Lead Fermi surface

Accurate description of Fermi surface properties requires a detailed sampling of the Brillouin Zone

Al Fermi surface

Representing a Fermi Surface

Lead Fermi surface

•Accurate description of Fermi surface properties requires a detailed sampling of the Brillouin Zone

• Full first-principles calculation expensive

Al Fermi surface

Scalar Relativistic with spin-orbit coupling Fe fermi surface

Outline

•Wannier Functions •One band

•Isolated Set of Bands

•Entangled Bands

•Wannier Interpolation•Accurate and Efficient approach to Fermi surface and spectral properties

•Examples•Anomalous Hall Effect

•Electron-Phonon Coupling

Wannier Functions

-9-8.5

-8-7.5

-7-6.5

-6-5.5

-5

L G X K G

wnR(r) =V

(2!)3

!

BZ"nk(r)e!ik.Rdk !nk(r)! e!i!(k)!nk(r)

Bloch stateQuantum # kWannier function

Quantum # R

GaAs

Wannier Functions

-9-8.5

-8-7.5

-7-6.5

-6-5.5

-5

L G X K G

-6-4-2 0 2 4 6

L G X K G

wnR(r) =V

(2!)3

!

BZ"nk(r)e!ik.Rdk !nk(r)! e!i!(k)!nk(r)

!nk(r)!!

n

U (k)mn!mk(r)

Multiband - Generalized WF

Bloch stateQuantum # kWannier function

Quantum # R

GaAs

Si

Maximally Localised Wannier Functions

wnR(r) =V

(2!)3

!

BZ

"#

m

U (k)mn"mk(r)

$e!k.Rdk

Choose Uk to minimise quadratic spread (Marzari, Vanderbilt)

|uk!

!uk|uk+b"U (k)Ab-initio code Wannier code

Wannier localisation in R gives Bloch smoothness in k !rot

nk (r) =!

m

U (k)mn!mk(r)

WF defined in basis of bloch states

-6-4-2 0 2 4 6

L G X K G

Si valence bands

Wannier Functions: A local picture

Si GaAs

1- Intuitive picture of local bonding

2- Wannier centres give bulk polarisation in a ferroelectric

Paraelectric Ferroelectric

BaTiO3

MLWF - Entangled Bands

Disentanglement procedure(Souza, Marzari and Vanderbilt)

Obtain “partially occupied” Wannier functions

Essentially perfect agreement within given energy window

Copper

Wannier InterpolationAccurate and Efficient approach to Fermi surface and spectral properties

•Exploit localisation of Wannier functions.•Require only matrix elements between close

neighbours.

Lead Fermi surface

1st principles accuracy at tight-binding costInterpolation of any one-electron operator

Wannier Interpolation

1- Obtain operator in Wannier basisOmn(R) = !Om|O|Rn" =

!

k

e!ik·R UO(k)U†

Wannier Interpolation

1- Obtain operator in Wannier basis

2- Fourier transform to an arbitrary k-point, k’Omn(k!) =

!

R

eik.·R Omn(R)

Omn(R) = !Om|O|Rn" =!

k

e!ik·R UO(k)U†

Wannier Interpolation

1- Obtain operator in Wannier basis

2- Fourier transform to an arbitrary k-point, k’

3- Un-rotate matrix at k’

Omn(k!) =!

R

eik.·R Omn(R)

!!m,k! |O|!n,k!" = [U†k!O(k!)Uk! ]

What we need diagonalises H

Omn(R) = !Om|O|Rn" =!

k

e!ik·R UO(k)U†

All operations involve small matrices (NwanxNwan) => FAST!

ConvergenceWannier interpolated object converges to ab-initio value, exponentially with k-grid sampling.

Lead

4x4x4

10x10x10 K-mesh size

Band interpolation error

Wannier Interpolation - Summary

Bloch Stateseg planewave basis

• Each k-point expensive• Calculation of MLWF

converges exponentially

Wannier basis

•Each k-point cheap•Introduce occupancies and compute properties (only polynomial convergence)

Sample on coarse grid Sample on fine grid

Wannier Interpolation - Fe•18 spinor Wannier functions•Keep up to 4th neighbour overlaps•Cost 1/2000 of full calculation

Wannier Interpolation

H !-2

-1.5

-1

-0.5

0

0.5

1

Wannier Interpolation pt2

1- k-derivatives can be taken analytically

eg band gradient

2- Position operator matrix elements well defined

Hk =!

R

eik·RH(R)

!Hk

!k!= i

!

R

R!eik·RH(R)

repeat for higher derivatives

Hall Effect!!"#$%&'#(!)'**!+,,+-.!/)'**!01234

!!5&67'*689!)'**!+,,+-.!/)'**!011:4!

!!;6!!!,%+*$!&++$+$<

!!=+##67'>&+.%97!?#+'@9!.%7+!#+A+#9'*

!

!!B#699+$!!"!'&$!!!!,%+*$9

!!!!?#+'@9!.%7+C#+A+#9'*

!!;6!7'>&+.%-!6#$+#!&++$+$

!

!!"#$%&'#(!)'**!+,,+-.!/)'**!01234

!!5&67'*689!)'**!+,,+-.!/)'**!011:4!

!!;6!!!,%+*$!&++$+$<

!!=+##67'>&+.%97!?#+'@9!.%7+!#+A+#9'*

!

!!B#699+$!!"!'&$!!!!,%+*$9

!!!!?#+'@9!.%7+C#+A+#9'*

!!;6!7'>&+.%-!6#$+#!&++$+$

!

Ordinary Hall Effect (Hall 1879)

Anomalous Hall Effect (Hall 1880)

Crossed E and B fields

B breaks time-reversal

No B field needed!

Ferromagnetism breaks time-reversal

Mechanisms for the AHEIntrinsic

1954 Karplus and Luttinger property of electron motion in perfect lattice “dissipationless” current (spin-orbit effect)

Extrinsic1955 Smit “skew scattering”1970 Berger “side-jump”

Intrinsic (again)1996- Rexamination of KL in terms of Berry’s phases Niu (Texas), Nagaosa (Tokyo)

Electron Dynamics

r =1!

!"nk

!k! k"!n(k)

!k = !eE! er"B

Textbook wavepacket dynamics

Electron Dynamics

r =1!

!"nk

!k! k"!n(k)

!k = !eE! er"B

Textbook wavepacket dynamics missing a term: “anomalous velocity”

“k-space magnetic field” caused by lattice potential“k-space Lorentz force” even when B=0

!xy =!e2

(2")2h

!

n

"

BZdk fn(k) !n,z(k)

Fermi function

Anomalous Hall Conductivity

Berry Curvature

Jonathan Yates Tyndall Institute 27th March 2007

Finite-differences difficult: phases, band-crossings

bcc iron (LAPW) Yao et al (PRL 2004)

Kubo formula

Extremely computationally expensive!final number within 20% of experiment

Berry Curvature

!n(k) = !Im"#kun,k|$ |#kun,k%

!n(k) = !!An(k)An(k) = i!unk|!k|unk"

!n,z(k) = !2Im!

m!=n

vnm,x(k) vmn,y(k)(!m(k)! !n(k))2

Berry Curvature

Berry Potential

Ab-initio Calculation

cell periodic Bloch state

Band-structure bcc Fe

N P! N H! N P H!

-8

-6

-4

-2

0

2

4

Spin up Spin downDFT (PBE) + spin-orbit

Berry curvature bcc Fe

-1e3

1e3

-1e5

1e5

1e1 -1e1

N P! N H! N P H!

-8

-6

-4

-2

0

2

4

Berr

y Cu

rvat

ure

(ato

mic

uni

ts)

Berry curvature bcc Fe

-1e3

1e3

-1e5

1e5

1e1 -1e1

N P! N H! N P H!

-8

-6

-4

-2

0

2

4

Berr

y Cu

rvat

ure

(ato

mic

uni

ts)

Berry Curvature

-1e3

1e3

-1e5

1e5

1e1 -1e1

P H!

-0.4

-0.2

0

0.2

0.4

Berr

y Cu

rvat

ure

(ato

mic

uni

ts)

!tot,z =!

n

fn(k) !n,z(k)

Total Berry Curvature

Our CalculationAb-initio k-grid: 8x8x8Interpolation k-grid: 320x320x320 + 7x7x7 “adaptive refinement”

(when Ω> 100 a.u.) Wannier Interpolation

planewaves + pseudopotentials

758 (Ω cm)-1

Existing Calculation*Kubo formula +

LAPW

751 (Ω cm)-1

*Yao et al PRL (2004)

Experiment ~1000 (Ω cm)-1

Differences: Scattering, Approximate DFT, Experimental uncertainty

Berry curvature in ky=0 plane

Total Berry curvature (atomic units)

Berry curvature in ky=0 plane

26% opposite spin

21% like spin

53% smooth background

Contributions

Total Berry curvature (atomic units)

Wannier Interpolation

•Map low energy electronic structure onto “exact tight binding model”

•Compute quantities with ab-initio accuracy and tight-binding cost

Basis set independent (planewaves, LAPW, guassians etc)Any single particle level of theory (LDA, LDA+U, GW)

•Many applications:Accurate DOS, joint-DOS, Fermi SurfacesSpin-relaxationMagneto-crystalline anisotropyNMR in metals (Knight shift)Electron-phonon interaction

AcknowledgmentsIvo Souza : UC Berkeley

Wannier Interpolation: Phys Rev B 75 195121 (2007)

Anomalous Hall EffectXinjie Wang, David Vanderbilt : Rutgers UniversityPhys Rev B 74 195118 (2006)

Wannier90 codeArash Mostofi, Nicola Marzari MITReleased under the GPL at www.wannier.org

Electron-PhononFeliciano Guistino, Marvin Cohen, Steven Louie : UCBPhys Rev Lett 94 047005 (2007)

Funding - Lawrence Berkeley National Laboratory Corpus Christi College, Cambridge