Wannier Function Based First Principles Method for Disordered

SystemsTom Berlijn, Wei Ku

PhDpostdoc

CMSN network

Collaborators

Wei Ku Dmitri

VoljaChi-Cheng

Lee

Chai-Hui

LinWei-Guo

Yin

Limin

Wang

William

Garber

Peter

HirschfeldYan

Wang

Theory

Theory

Andrivo

Rusydi

Tun Seng

Herng

Dong

Chen Qi

Experiment

Ding

Jun

Outline

• Introduction: Super Cell Approximation

• Method : Effective Hamiltonian (Wannier function) [1]

• Application 1: eg’ pockets of NaxCoO2 [1]

• Application 2: oxygen vacancies in Zn1-xCuxO1-y[2]

[1] T. Berlijn, D. Volja, W. Ku, PRL 106 077005 (2011)

[2] T. S. Herng, D.-C. Qi, T. Berlijn, W. Ku, A. Rusydi et al, PRL 105, 207201 (2010)

Introduction:

Super Cell Approximation

What kind of disorder?Not like

But like

substitution interstitial vacancy

Goal: configurationally averaged spectral function of disordered systems from first principles

mean field vs super cell

<A(k,w) > = S A(k,w)iconfig i

Mean Field1

VCA: Virtual Crystal Approximation

CPA: Coherent Potential Approximation

no scattering

non-local physics missing1) A. Gonis, “Green functions for ordered and disordered systems” (1992)

Vvirtual crystal = (1-x) VA + x VB

non-local physics 1: k-dependent self-energy S(w,k)

w

kbut mean-field k-independent S(w)

non-local physics 2 : Large-sized impurity states (Anderson localization)

non-local physics 3 : Short Range Order

Approach: super cell approximation

problem

band folding

computational expense

solution

unfolding[1]

effective Hamiltonian[1] W. Ku , T. Berlijn, and C.-C. Lee, PRL 104 216401 (2010)

<A(k,w) > ≈ 1/N ( A1(k,w) + … + AN(k,w) )

Method:

Effective Hamiltonian

T. Berlijn, D. Volja, W. Ku, PRL 106 077005 (2011)

H = H0 + Si D(i) + Si,j D(i,j) + …

D(i) = H(i) - H0

D(i,j) = H(i,j) - D(i) - D(j) - H0

higher order moments

Concept: Linearityundoped linear 2-bodydisordered

Construction

1. DFT doped & undoped

2. Wannier-transformation

3. Linear superposition

1) Density Functional Theory

undoped

(normal cell)

1 impurity

(per super cell)

two DFT Calculations

Influence impurity

2) Wannier transfomation

DFT 1 impurity

2 Wannier transformations

2 Tight Binding Hamiltonians

undoped 1 impurity

DFT undoped

En

erg

yE

ner

gy

k

K

Hdft0 Hdft

(i)

|rn> = e-ik•r Unj(k) |kj>kj

3) Linear Superposition

Influence 1 impurity:

D(i) = Hdft(i) - Hdft

0

effective Hamiltonian N impurities:

Heff(1,…,N) = Hdft

0 + SiD(i)

Testing DFT v.s. effective Hamiltonian

x=0

x=2/3

x=1/8

Test : NaxCoO2

Effective HamiltonianDFTE

ne

rgy

(e

V)

Test : NaxCoO2

66 Wanier Functions

1 diagonalization

2019 LAPW’s + 164 LO’s

self consistency

Co-eg

Co-ag

Co-eg’

O-p

x=0

x=1

Test Zn1-xCuxO (rock salt)

x=1/4

ZnO CuO

Zn-d

Cu-d O-p hybrid

Cu-d

-8

-4

0

4

8

-8

-4

0

4

8

En

erg

y (

eV

)

Effective HamiltonianDFT

Test : Zn1-xCuxO (rock salt)

spin spin

spin spin

-8

-4

0

4

8

-8

-4

0

4

8

x=0

x=1/8

Test Zn1-xCuxO (rock salt)

x=1/4

-8

-4

0

4

8

-8

-4

0

4

8

-8

-4

0

4

8

-8

-4

0

4

8

En

erg

y (

eV

)

Effective HamiltonianDFT

Test : Zn1-xCuxO (rock salt)

spin spin

spin spin

Application 1:

eg’ pockets of NaxCoO2

T. Berlijn, D. Volja, W. Ku, PRL 106 077005 (2011)

Why NaxCoO2?High Thermoelectric

Power1

Unconventional Super Conductivity2 ?

1) I. Terasaki et al, PRB 56 R12 685 (1997)

2) K. Takada et al, nature 422 53 (2003)

Intercalation: NaxCoO2

Q) Does Na disorder destroy eg’ pockets3 ?

1) D.J. Singh, PRB 20, 13397 (2000)

2) D. Qian et al, PRL 97 186405 (2006)

3) David J. Singh et al, PRL 97, 016404-1 (2006)

LDA1 ARPES2

eg’

ag

configuration 1 configuration 50

+ . . . +

NaxCO2 : x0.30

50 configurations of ~200 atoms

En

erg

y (

eV

)

Co-ag

-0.20.00.2

-0.20.00.2

-4.2-4.0-3.8

0.20.1

Co-eg’

O-p

A) Na disorder does not destroy eg’ 1

En

erg

y (

eV

)non-local physics :

k-dependent broadening

0.2

0.1

0.0

-0.1H A G K

Co-ag

Co-eg’

k-dependent self energy S(k,w)?

non-local physics : short range order

Na(1) above Co

Na(2) above Co-hole

Simple rule: Na(1) can

not sit next to Na(2):

non-local physics : short range order

0.2

0.1

0.0

-0.1H A G K 0.0 0.2 0.4 0.6

A(k,w) A(k0,w) @ k0=G

Ene

rgy

(eV

)

X=0.30

0.2

0.1

0.0

-0.1H A G K 0.0 0.2 0.4 0.6

A(k,w) A(k0,w) @ k0=GE

nerg

y (e

V)

X=0.70

0.3 0.2 0.1 0.0-0.1-0.2

0.3 0.2 0.1 0.0-0.1-0.2

SRO suppresses ag broadening?

Na(1) islandNa(2) island

Application 2:

oxygen vacancies in Zn1-xCuxO1-y

T. S. Herng, D.-C. Qi, T. Berlijn, W. Ku, A. Rusydi et al, PRL 105, 207201 (2010)

Experiment

First principles simulation

Microscopic picture

• Film growth & charactarizationDr. T. S. Herng (NUS)Prof. Ding Jun (NUS)

• Beamline scientistsDr. Gao Xingyu (NUS)Dr. Yu Xiaojiang (SSLS)Cecilia Sanchez-Hanke

(NSLS)

• XAS & XMCDDr. Qi Dongchen (NUS)Prof. A. Rusydi (NUS)

Tom BerlijnWei Ku

three representative films

1. ZnO

2. Cu:ZnO O-rich (2% Cu)

3. Cu:ZnO O-poor (2% Cu & ~1% oxygen vacancies VO)

SQUID

ZnO @ 300K

ZnO:Cu @ 300K (O-rich)

ZnO:Cu @ 5K (O-poor)

ZnO:Cu @ 300K (O-poor)

Observation: oxygen vacancy induce FM @ 300K in Cu:ZnO

NB: 2 Cu-d9 + Vo = 2 Cu-d10

Q: what is the role of oxygen vacancies?

Q What is the influence of the oxygen vacancies?

oxygen vacancy = attractive potential + 2 electrons

Q ) Where do the electrons go? A) one-particle spectral function <A(k,w)>

of Zn1-xCuxO1-y with attractive potential VO but without its donated electrons

Zn Cu↓ Cu↑ O VO

configurationally averaged spectral function

<A(k,w) > ≈ 1/10 ( A1(k,w) + … + A10(k,w) )

configuration 1 configuration 10

≈1/10 +….+

spectral function <A(k,w)>

conduction bandZn-4s

Cu-3d↑

valence band O-2p Cu-3d↓

VO

Cu-3d x 40

<A↑(k,w)> <DOS↑(w)> <DOS↓(w)> <A↓(k,w)>

Ene

rgy

(eV

)

Q) Where do the electrons go?

A) Cu upper Hubbard level(leaving VO empty)

Ene

rgy

(eV

)

<A↑(k,w)> <DOS↑(w)>

Cu-3d↑

VO

Zn-4s

O-2p

e- Cu-3d x 40

Ene

rgy

(eV

)

Cu-3d x 40

<A↑(k,w>

Cu-3d↑VO

Zn-4s

O-2p

Oxygen vacancy states |VO> are big

k-space real-space

FWHM ≈GM/5 oxygen vacancy

wavefunction <x|Vo>

radius ≈ 2.5 a

Ene

rgy

(eV

)

Cu-3d x 40

<A↑(k,w>

Cu-3d↑VO

Zn-4s

O-2p

But why are oxygen vacancy states |VO> so big?

k-space real-space

Attractive potential

|Vo> ≈ ½ (|Zn1-s> +|Zn2-s> +|Zn3-s> +|Zn4-s>)

Oxygen vacancy = attractive potential + 2 electrons

Attractive potential in the 4 neighboring Zn

Zn

O

1. Electrons go into |Cu-d↑>2. Oxygen vacancy states |VO> are big

First principles results

Microscopic picture

no vacancies with vacancies

Conclusion: oxygen vacancies mediate the Cu moments

VOCu d 9 Cu d 10

Outline

• Introduction: Super Cell Approximation

• Method : Effective Hamiltonian (Wannier function) [1]

• Application 1: eg’ pockets of NaxCoO2 [1]

• Application 2: oxygen vacancies in Zn1-xCuxO1-y[2]

[1] T. Berlijn, D. Volja, W. Ku, PRL 106 077005 (2011)

[2] T. S. Herng, D.-C. Qi, T. Berlijn, W. Ku, A. Rusydi et al, PRL 105, 207201 (2010)