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Wannier Function Based First
Principles Method for Disordered
Systems
Tom Berlijn, Wei Ku
postdoc
PhD
CMSN
network
Collaborators
Theory
Wei Ku
Dmitri Chi-Cheng Chai-Hui
Wei-Guo
Volja
Lee
Lin
Yin
Limin William
Wang Garber
Experiment
Andrivo Tun Seng Dong Ding
Rusydi Herng Chen Qi Jun
Theory
Peter
Hirschfeld
Yan
Wang
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
<A(k,w) > =
S
config i
A(k,w)i
mean field vs super cell
Mean Field1
VCA: Virtual Crystal Approximation
Vvirtual crystal =
(1-x) VA + x VB
no scattering
CPA: Coherent Potential Approximation
non-local physics missing
1) A. Gonis, “Green functions for ordered and disordered systems” (1992)
non-local physics 1:
k-dependent self-energy S(w,k)
w
but mean-field k-independent S(w)
k
non-local physics 2 : Large-sized impurity
states (Anderson localization)
non-local physics 3 : Short Range Order
Approach: super cell approximation
<A(k,w) > ≈ 1/N (
A1(k,w) + … +
problem
band folding
computational expense
AN(k,w) )
solution
unfolding[1]
effective Hamiltonian
[1] W. Ku , T. Berlijn, and C.-C. Lee, PRL 104 216401 (2010)
Method:
Effective Hamiltonian
T. Berlijn, D. Volja, W. Ku, PRL 106 077005 (2011)
Concept: Linearity
disordered
undoped
2-body
linear
H = H0 + Si D(i) + Si,j D(i,j) + …
D(i) = H(i) - H0
(i,j)
D =
(i,j)
H -
(i)
D -
(j)
D -
0
H
higher order moments
Construction
1. DFT doped & undoped
2. Wannier-transformation
3. Linear superposition
1) Density Functional Theory
two DFT Calculations
undoped
(normal cell)
1 impurity
(per super cell)
Influence impurity
2) Wannier transfomation
DFT undoped
2 Wannier transformations
Energy
|rn> =S e-ik•r Unj(k) |kj>
kj
k
DFT 1 impurity
Energy
2 Tight Binding Hamiltonians
undoped
0
Hdft
K
1 impurity
(i)
Hdft
3) Linear Superposition
Influence 1 impurity:
D(i) = Hdft(i) - Hdft0
effective Hamiltonian N impurities:
Heff(1,…,N) = Hdft0 + SiD(i)
Testing
DFT v.s. effective Hamiltonian
Test : NaxCoO2
x=0
x=2/3
x=1/8
Effective Hamiltonian
Energy (eV)
DFT
Test : NaxCoO2
Co-eg
Co-ag
Co-eg’
O-p
2019 LAPW’s + 164 LO’s
66 Wanier Functions
self consistency
1 diagonalization
Test Zn1-xCuxO (rock salt)
x=0
x=1
x=1/4
ZnO
CuO
Cu-d O-p hybrid
Cu-d
Zn-d
Test : Zn1-xCuxO (rock salt)
DFT
Energy (eV)
8
spin
Effective Hamiltonian
8
4
4
0
0
-4
-4
-8
-8
8
spin
8
4
4
0
0
-4
-4
-8
-8
spin
spin
Test Zn1-xCuxO (rock salt)
x=0
x=1/8
x=1/4
Test : Zn1-xCuxO (rock salt)
DFT
Energy (eV)
8
spin
Effective Hamiltonian
8
4
4
0
0
-4
-4
-8
-8
8
spin
8
4
4
0
0
-4
-4
-8
-8
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
LDA1
ARPES2
ag
eg’
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)
NaxCO2 : x0.30
50 configurations of ~200 atoms
configuration 1
configuration 50
+...+
Energy (eV)
0.2
0.0
-0.2
0.2
0.0
-0.2
-3.8
-4.0
-4.2
Co-eg’
Co-ag
O-p
0.1 0.2
A) Na disorder does not destroy eg’ 1
Energy (eV)
non-local physics :
k-dependent broadening
Co-eg’
0.2
0.1
Co-ag
0.0
-0.1
H
AG
K
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
A(k0,w) @ k0=G
Energy (eV)
A(k,w)
X=0.30
0.2
0.1
0.0
-0.1
H
X=0.70
Na(1) island
Energy (eV)
Na(2) island
0.3
0.2
0.1
0.0
-0.1
-0.2
A G
K
0.0 0.2 0.4 0.6
A(k,w)
0.3
0.2
0.1
0.0
-0.1
-0.2
0.2
0.1
0.0
-0.1
H
A G
K
A(k0,w) @ k0=G
0.0 0.2 0.4 0.6
SRO suppresses ag broadening?
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)
• Film growth & charactarization
• XAS & XMCD
Dr. T. S. Herng (NUS)
Dr. Qi Dongchen (NUS)
Prof. Ding Jun (NUS)
Prof. A. Rusydi (NUS)
• Beamline scientists
Dr. Gao Xingyu (NUS)
Dr. Yu Xiaojiang (SSLS)
Cecilia Sanchez-Hanke (NSLS)
Experiment
Microscopic
picture
Wei Ku
First principles
simulation
Tom Berlijn
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
configurationally averaged spectral function
<A(k,w) > ≈ 1/10 (
A1(k,w) + … +
configuration 1
≈1/10
Zn
A10(k,w) )
configuration 10
+….+
Cu↓
Cu↑
O
VO
spectral function <A(k,w)>
<A↑(k,w)>
<DOS↑(w)> <DOS↓(w)>
Energy (eV)
conduction
band
Zn-4s
Cu-3d↑
valence band
O-2p
Cu-3d x 40
VO
Cu-3d↓
<A↓(k,w)>
Q) Where do the electrons go?
Energy (eV)
<A↑(k,w)>
e-
<DOS↑(w)>
Zn-4s
Cu-3d x 40
VO
Cu-3d↑
O-2p
A) Cu upper Hubbard level
(leaving VO empty)
Oxygen vacancy states |VO> are
big
k-space
real-space
Energy (eV)
<A↑(k,w>
FWHM ≈GM/5
Zn-4s
oxygen vacancy
wavefunction <x|Vo>
VO
Cu-3d↑
O-2p
Cu-3d x 40
radius
≈ 2.5 a
But why are oxygen vacancy states
|VO> so big?
real-space
Zn
<A↑(k,w>
Energy (eV)
Oxygen vacancy = attractive
potential + 2 electrons
k-space
Attractive
potential
VO
Zn4s
Cu-3d↑
O2p
O
Attractive potential in the 4
neighboring Zn
Cu-3d x 40
|Vo> ≈ ½ (|Zn1-s> +|Zn2-s> +|Zn3-s> +|Zn4-s>)
First principles results
1. Electrons go into |Cu-d↑>
2. Oxygen vacancy states |VO> are big
Microscopic picture
with vacancies
no vacancies
Cu d 9
Cu d 10
VO
Conclusion: oxygen vacancies mediate the Cu moments
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)

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