the all-electron gw method based on wien2k: implementation and applications

The all-electron GW method based on WIEN2k:Implementation and applications.

Ricardo I. Gomez-Abal

Fritz-Haber-Institut of the Max-Planck-SocietyFaradayweg 4-6, D-14195, Berlin, Germany

15th. WIEN2k-WorkshopMarch, 29th. 2008

R. Gomez-Abal (FHI-Berlin) GW@Wien2k Vienna 2008 1 / 54

Outline

Outline

1 Introduction

2 ImplementationGW@wien2kConvergence Tests

3 ResultsBandgapsBandstructuresMacroscopic Dielectric Constant

4 Core-valence interactionBandgapsSemicore States

5 f-electron systems

6 Conclusions


Introduction

Density Functional Theory

E ⇐⇒ ρ

Kohn-Sham scheme

interacting electrons fictious particles

non interacting

Condition:

n(r) =

occ∑

i

|Ψi(r)|2


Introduction

Density Functional Theory

Kohn-Sham equation

[T + Vext + VH + Vxc ]Ψi = ǫiΨi

Vxc

localenergy independenthermitian

ǫi :

Lagrange multipliersNo physical meaningException: Highest occupied ǫi = −I

Fast.

Good structural properties.

Excitation spectra??

E ⇐⇒ ρR. Gomez-Abal (FHI-Berlin) GW@Wien2k Vienna 2008 4 / 54

Introduction

The Bandgap Problem

LDA vs. experimental bandgaps

Si

GaA

s

zGaN

ZnS

C

CaO M

gO

NaC

l

0 2 4 6 8Eg

exp [eV]

0

2

4

6

8E

gLDA [

eV]

Up to 50% underestimationR. Gomez-Abal (FHI-Berlin) GW@Wien2k Vienna 2008 5 / 54

Introduction

Many-Body Theory

Quasiparticle equation

[T + Vext + VH ]Ψi (r) +

∫

Σ(r, r′; ǫi )Ψi (r′)d3r′ = ǫiΨi(r)

Σ

non localenergy dependentnon hermitian

ǫi :

Poles of the Green’s Function

ǫi =

E (N) − E (N − 1, i) ǫi < EF

E (N + 1, i) − E (N) ǫi > EF

Formally correspond to the excitation spectra


Introduction

Many Body Theory (cont.)

The Self-Energy (Σ)

Hedin, 1965: Expansion in terms of the dynamically screened Coulombpotential (W ).

Fast convergence.

First order:

Σ(r, t, r′, t ′) = G (r, t, r′, t ′)W (r, t, r′, t ′)

Simplest approximation including dynamical correlation effects


Introduction

Many Body Theory (cont.)

The Screened Coulomb potential (W )

W (r1, r2;ω) =

∫

ε−1(r1, r3;ω)v(r3, r2)dr3

ε(r1, r2;ω) =1 −∫

v(r1, r3)P(r3, r2;ω)dr3

P(r1, r2;ω) = − i

2π

∫

G (r1, r2;ω + ω′)G (r2, r1;ω‘)dω′

Requires selfconsistency with the QP equation!


Introduction

Perturbative treatment

G0W0

[T + Vext + VH ]Ψi (r)+∫

[

Σ(r, r′; ǫi)]

Ψi(r′)d3r′ = ǫiΨi(r)


Introduction


G0W0

[T + Vext + VH + Vxc ]Ψi (r)+∫

[

Σ(r, r′; ǫi ) − Vxc(r′)δ(r − r′)

]



Introduction


G0W0


[

Σ(r, r′; ǫi ) − Vxc(r′)δ(r − r′)

]


First order correction to ǫKSnk :

ǫqpnk = ǫKS

nk + ∆ǫnk

∆ǫnk = ℜ(〈Ψnk(r)|Σ(r, r′, ǫqpnk)|Ψnk(r

′)〉) − 〈Ψnk(r)|Vxc |Ψnk(r)〉


Introduction


G0W0


[

Σ(r, r′; ǫi ) − Vxc(r′)δ(r − r′)

]


First order correction to ǫKSnk :

ǫqpnk = ǫKS

nk + ∆ǫnk

∆ǫnk = ℜ(〈Ψnk(r)|Σ(r, r′, ǫqpnk)|Ψnk(r

′)〉) − 〈Ψnk(r)|Vxc |Ψnk(r)〉Σ calculated in the GW approximation.


Introduction

Introduction

G0W0 equations

G0(r1, r2;ω) =occ∑

nk

Ψnk(r1)Ψ∗nk(r2)

ω − ǫnk − iη+

unocc∑

nk

Ψnk(r1)Ψ∗nk(r2)

ω − ǫnk + iη

P(r1, r2;ω) = − i

2π

∫

G0(r1, r2;ω + ω′)G0(r2, r1;ω‘)dω′

ε(r1, r2;ω) =1 −∫

v(r1, r3)P(r3, r2;ω)dr3

W0(r1, r2;ω) =

∫

ε−1(r1, r3;ω)v(r3, r2)dr3

Σ(r1, r2;ω) =i

2π

∫

G0(r1, r2;ω + ω′)W0(r2, r1;ω‘)dω′


Implementation GW@wien2k

Implementation


Implementation Basis Functions

The Polarization

P(r1, r2, ω) =X

n,m,k,k′

Ψnk(r1)Ψ∗

mk′(r1)Ψ∗

nk(r2)Ψmk′(r2)F(ǫnk, ǫmk′ ; ω)



The basis functions

P(r1, r2, ω) =X

n,m,k,k′

Ψnk(r1)Ψ∗

mk′(r1)Ψ∗

nk(r2)Ψmk′(r2)F(ǫnk, ǫmk′ ; ω)

Definition

χqi (r) =

∑

Ra

e ik·(R+ra)υNL(r)YLM(r) r ∈ MT-spheres

1√V

∑

G

Si ,Ge i(q+G)·r r ∈ Interstitial

1 F. Aryasetiawan and O. Gunnarsson, Rep. Prog. Phys. 61, 237 (1998).

2 T. Kotani and M. van Schilfgaarde, Sol. State. Comm. 121, 461 (2002).

MTSphere

Interstitial



The basis functions

Obtaining the radial functions

For each L take ul(r)ul ′(r) such that |l − l ′| ≤ L ≤ l + l ′

Calculate the overlap matrix:

Oall ′,l1l

′

1=

RMT∫

0

ul(r)ul ′(r)ul1(r)ul ′1(r)r2dr

Solve the secular equation:

(

Oall ′,l1l

′

1− λnδll1δl ′l ′1

)

cn,l1l′

1= 0

If λn ≥ λtol then:

υnL(r) =∑

ll ′

cn,ll ′ul (r)ul ′(r)



The basis functions

The matrix elements

M inm(k,q) ≡

∫

Ω

[

χqi (r)Ψm,k−q(r)

]∗Ψnk(r)d

3r

Tests

Visual

Ψn,k(r)Ψ∗m,k−q(r) =

∑

i

M inm(k,q)χq

i (r)

Completeness

∆ =

R

|Ψn,k(r)Ψ∗

m,k−q(r)−

P

i Minm(k,q)χq

i(r)|2d3r

R

|Ψn,k(r)Ψ∗

m,k−q(r)|2d3r

∆ = 1 −∑

i |M inm(k,q)|2

∫

|Ψn,k(r)Ψ∗m,k−q(r)|2d3r



Basis Functions: Tests

k = Γ, k′ = X , n = 4 m = 5

At1 RMT1RMT2

At2r

-0.04

-0.02

0

0.02

Ψnk

Ψm

k’ExactFit

Ψn,k(r)Ψ∗

m,k−q(r) =X

i

M inm(k, q)χq

i(r)




k = Γ, k′ = π

a(1, 0, 0), n = 2 m = 6

At1 RMT1RMT2

At2r

-0.01

0

0.01Ψ

nkΨ

mk’

ExactFit

Ψn,k(r)Ψ∗

m,k−q(r) =X

i

M inm(k, q)χq

i(r)




k = 3π

2a(1, 1,−1), k′ = π

2a(1,−1, 1), n = 2, m =

6

At1 RMT1RMT2

At2r

-0.1

-0.05

0

0.05

0.1Ψ

nkΨ

mk’

ExactFit

Ψn,k(r)Ψ∗

m,k−q(r) =X

i

M inm(k, q)χq

i(r)




k = Γ, k′ = π

a(1, 0, 0), n = 1, m = 5

At1 RMT1RMT2

At2r

0

0.5

1Ψ

nkΨ

mk’

ExactFit

Ψn,k(r)Ψ∗

m,k−q(r) =X

i

M inm(k, q)χq

i(r)



Basis Functions: Completeness test.

MT-Spheres

20 40 60 80 100 120 140 160 180

Number of spherical basis functions per atom

1e-06

1e-05

1e-04

0.001

0.01

0.1

Rel

ativ

e er

ror

tol.

= 1

e-2

tol.

= 1

e-3

tol.

= 1

e-4

tol.

= 1

e-5

tol.

= 1

e-7

tol.

= 1

e-2

tol.

= 1

e-3

tol.

= 1

e-4

tol.

= 1

e-5

lmax=2 lmax=3lmax=1

tol.

= 1

e-2

tol.

= 1

e-3

tol.

= 1

e-5

MinimumMaximum




MT-Spheres

20 40 60 80 100 120 140 160 180


1e-06

1e-05

1e-04

0.001

0.01

0.1

Rel

ativ

e er

ror

tol.

= 1

e-2

tol.

= 1

e-3

tol.

= 1

e-4

tol.

= 1

e-5

tol.

= 1

e-7

tol.

= 1

e-2

tol.

= 1

e-3

tol.

= 1

e-4

tol.

= 1

e-5

lmax=2 lmax=3lmax=1

tol.

= 1

e-2

tol.

= 1

e-3

tol.

= 1

e-5

MinimumMaximum

Remarks

Decreases with lmax .




MT-Spheres

20 40 60 80 100 120 140 160 180


1e-06

1e-05

1e-04

0.001

0.01

0.1

Rel

ativ

e er

ror

tol.

= 1

e-2

tol.

= 1

e-3

tol.

= 1

e-4

tol.

= 1

e-5

tol.

= 1

e-7

tol.

= 1

e-2

tol.

= 1

e-3

tol.

= 1

e-4

tol.

= 1

e-5

lmax=2 lmax=3lmax=1

tol.

= 1

e-2

tol.

= 1

e-3

tol.

= 1

e-5

MinimumMaximum

Remarks


Saturates with ǫtol

for a given lmax .




Interstitial

150 200 250 300 350

Number of interstitial basis functions

1e-08

1e-07

1e-06

1e-05

1e-04

0.001

Rel

ativ

e er

ror

LAP

W b

asis

MinimumMaximum

Remarks


Saturates with ǫtol fora given lmax .

Decreases with

|Gmax |.




Interstitial

150 200 250 300 350

Number of interstitial basis functions

1e-08

1e-07

1e-06

1e-05

1e-04

0.001

Rel

ativ

e er

ror

LAP

W b

asis

MinimumMaximum

Remarks


Saturates with ǫtol fora given lmax .

Decreases with |Gmax |.

Recipe

Choose max(ǫtol ) thatsaturates

Choose |Gmax | so that∆i ≈ ∆MT


Implementation Brillouin Zone Integration

The Polarization

Pij(q, ω) =

Z

BZ

X

nm

M inm(k,q)[M j

nm(k,q)]∗

ω−ǫmk−q+ǫnk+iη−

M inm(k,q)[M j

nm(k,q)]∗

ω−ǫnk+ǫmk−q−iη

ff

f (ǫk)[1 − f (ǫk−q)]d3k



Brillouin Zone Integration

Pij(q, ω) =

Z

BZ

X

nm

M inm(k,q)[M j

nm(k,q)]∗


M inm(k,q)[M j

nm(k,q)]∗


ff

f (ǫk)[1 − f (ǫk−q)]d3k

q-dependent Linear Tetrahedron Method




Pij(q, ω) =

Z

BZ

X

nm

M inm(k,q)[M j

nm(k,q)]∗


M inm(k,q)[M j

nm(k,q)]∗


ff

f (ǫk)[1 − f (ǫk−q)]d3k


Linear Tetrahedron Method

P(q, ω) =

Z

BZ

X (k, q)f (ǫk)[1 − f (ǫk−q)]

ω − ∆ǫd

3k

wTki ,q

=

Z

ΩT

wi (k)f (ǫk)[1 − f (ǫk−q)]

ω − ∆ǫd

3k wki ,q =

X

T∋ki

wTki ,q

P(q, ω) =X

nm

X

i

X (ki , q)wnm(ki , q; ω)

J. Rath and A. J. Freeman, Phys. Rev. B 11, 2109 (1975)




Pij(q, ω) =

Z

BZ

X

nm

M inm(k,q)[M j

nm(k,q)]∗


M inm(k,q)[M j

nm(k,q)]∗


ff

f (ǫk)[1 − f (ǫk−q)]d3k





Pij(q, ω) =

Z

BZ

X

nm

M inm(k,q)[M j

nm(k,q)]∗


M inm(k,q)[M j

nm(k,q)]∗


ff

f (ǫk)[1 − f (ǫk−q)]d3k


~q




Pij(q, ω) =

Z

BZ

X

nm

M inm(k,q)[M j

nm(k,q)]∗


M inm(k,q)[M j

nm(k,q)]∗


ff

f (ǫk)[1 − f (ǫk−q)]d3k




Brillouin Zone integration

Pij(q, ω) =

Z

BZ

X

nm

M inm(k,q)[M j

nm(k,q)]∗


M inm(k,q)[M j

nm(k,q)]∗


ff

f (ǫk)[1 − f (ǫk−q)]d3k


4 nodes 6 nodes 8 nodes 8 nodes




Test: Free electron gas

0 0.5 1 1.5 2 2.5 3 3.5q/kF

0

0.2

0.4

0.6

0.8

1

Lind

hard

t fun

ctio

n

exact364 k-pts540 k-pts1368 k-pts




Test: Cu bandstructure

W L Γ X Z W K

-10

EF

10 LDAGW




Test: Cu DOS

-8 -6 -4 -2 EF2 4

Energy [eV]

0

1

2

3

4

5

6D

OS

LDAGW




Test: Al bandstructure

W L Γ X Z W K

-10

EF

10

LDAGW


Implementation The Γ-point singularity

The Γ point singularity

The symmetrized dielectric matrix

Definition

εij (q, ω) =∑

lm

v12il (q)Plm(q, ω)v

12mj (q)

The screened potential

Wij(q, ω) =∑

lm

v12il (q)ε−1

lm (q, ω)v12mj (q)




The screened Coulomb potential

v12

ij (q → 0) =v

s 12

ij

|q|+ v

12

ij (q)

Wij(q, ω) =1

|q|2W s2

ij (q, ω) +1

|q|W s1

ij (q, ω) + Wij(q, ω)

Diverges!




The screened Coulomb potential

v12

ij (q → 0) =v

s 12

ij

|q|+ v

12

ij (q)

Wij(q, ω) =1

|q|2W s2

ij (q, ω) +1

|q|W s1

ij (q, ω) + Wij(q, ω)

But can be integrated


Implementation Frequency convolution

Implementation

Frequency convolution

Σ(r1, r2; ω) =i

2π

Z

G0(r1, r2; ω + ω′)W0(r2, r1; ω‘)dω

′

⇓ Analytic continuation

Σ(r1, r2; iω) =i

2π

Z

G0(r1, r2; iω + iω′)W0(r2, r1; iω‘)diω

′



Implementation


Σ(r1, r2; ω) =i

2π

Z

G0(r1, r2; ω + ω′)W0(r2, r1; ω‘)dω

′


Σ(r1, r2; iω) =i

2π

Z


′

Σnk(iω) = −X

q

X

ij

X

n′

M inn′ (k, q)

1

π

∞Z

0

Wij (q; iω′)dω′

(iω − ǫn,k′)2 + ω′2M

∗j

n′n(k, q)



Implementation


Σ(r1, r2; ω) =i

2π

Z

G0(r1, r2; ω + ω′)W0(r2, r1; ω‘)dω

′


Σ(r1, r2; iω) =i

2π

Z


′

Σnk(iω) = −X

q

X

ij

X

n′

M inn′ (k, q)

1

π

∞Z

0

Wij (q; iω′)dω′

(iω − ǫn,k′)2 + ω′2M

∗j

n′n(k, q)

Pade Approximant

Σnk(iω) =

PN

j=0 aj(iω)j

PN+1j=0 bj(iω)j


Σnk(ω) =

PN

j=0 ajωj

PN+1j=0 bjω

j


Implementation Summary

G0W0@Wien2k: A FP-(L)APW+lo + G0W0 code

Flowchart

Wien2k Begin

ψkn χiq

ǫDFTkn M i

nm(k, q) vij (q)

Pij (q, ω)

εij (q, ω)

Wij (q, ω)

V xck,n Σnn(k, ω)

ǫqpk,n

End

Code keywords

Based on FP-(L)APW+lo

Mixed Basis

Linear TetrahedronMethod

Reciprocal space

Imaginary frequencies

Excelent results


Implementation Convergence Tests

Silicon: Convergence tests

Number of frequencies

10 15 20 25Nr. of frequencies

0.878

0.879

0.88

0.881

0.882

0.883

0.884

Eg [e

V]

Used paremeters

16 frequencies




Number of excited states

8 27 64 125Nr. of k-points

0.45

0.46

0.47

0.48

0.49

0.5

0.51

0.52

0.53

∆Eg [e

V]

Used paremeters

16 frequencies

64 k-points




Number of k-points

0 50 100 150 200 250Number of unoccupied bands

0.95

1

1.05

1.1

Ban

d G

ap

Used paremeters

16 frequencies

64 k-points

∼ 200 unocc. bands


Results Bandgaps

Results I:

BandgapsS

i GaA

s

zGaN Z

nS

C

MgO

NaC

l

CaO

0 2 4 6 8Experimental Eg [eV]

0

2

4

6

8

10

Cal

cula

ted

Eg [e

V]

G0W0 All electronG0W0 PseudopotentialsG0W0@Wien2kLDA

- F. Aryasetiawan and O.Gunnarson, Rep. Prog. Phys.61, 237 (1998).(and Refs.)

- T. Kotani and M. vanSchilfgaarde, Solid State Comm.121, 461 (2002).

- C. Friedrich et al, Phys. Rev. B74, 045104 (2006).


Results Bandstructures

Results II:

Band diagrams: Silicon

W ΓL Λ ∆ X Z W K

-10

-5

εF

5

10

1

LDAGW


Core-valence interaction Motivation

Motivation

G0W0 Si bandgap

1990 2000Publication year

0

0.5

1

1.5

Eg [e

V]

PseudopotentialsAll electron


Core-valence interaction Motivation

Debate

G0W0 Si bandgap

1990 2000 2010Publication year

0

0.5

1

1.5

Eg [e

V]

PseudopotentialsAll electron

?

- M. Tiago et al, Phys. Rev. B 69, 125212 (2004).

- C. Friedrich et al, Phys. Rev. B 74, 045104 (2006).

Pseudopotentials

systematically largergaps

in better agreementwith expeeriment

All electron

Benchmark for ab-initio

calculations

- A. Fleszar, Phys. Rev. B 64, 245204(2001).

- T. Kotani and M. van Schilfgaarde,Solid State Comm. 121, 461 (2002).

- W. Ku and A. Eguiluz, Phys. Rev. Lett.89,126401 (2002).


Core-valence interaction Core-valence Partitioning

Core-valence partitioning

Kohn-Sham equation

HKSΨnk = ǫnkΨnk

All electron

HKS = T + Vnuc + Vh + Vxc [n]

Pseudopotentials

HKS = T + Vps + Vh + Vxc [nv + nc ]

Vxc [n] = Vxc [nv + nc ] + Vxc [ncore − nc ] ⊂ Veff

G0W0 correctionǫqpnk = ǫKS

nk + ∆ǫnk

All electron

∆ǫnk = ℜ (Σnk [Ψnk; Ψc])− V xcnk [n]

Pseudopotentials

∆ǫnk = ℜ(Σnk[Ψnk])−V xcnk [nv +nc ]

ℜ (Σnk [Ψnk; Ψc]) = ℜ(Σnk[Ψnk]) + V xcnk [ncore − nc ] ⊂ Veff


Core-valence interaction Different Approaches

Different Approaches

G0W0 correction

All electron:

ǫqpnk = ǫKS

nk + ℜ (Σnk [Ψnk; Ψcore]) − V xcnk [n]

Pseudopotentials:

ǫqpnk = ǫKS

nk + ℜ(Σnk[Ψnk]) − V xcnk [nval ]

Valence only:

ǫqpnk = ǫKS

nk + ℜ (Σnk [Ψnk]) − V xcnk [nval ]

Separated terms

Σnk = Σxnk + Σc

nk


Core-valence interaction Silicon

Results

Si: Matrix elements

Highest occ. state at Γ

-15

-10

-5

0

5

0.93

-12.95 -11.25

1.03

-12.97 -11.43

1.05

-14.95 -13.54

Psedopotential Valence only All electron

C

XVXC

Lowest unocc. state at X

-15

-10

-5

0 -3.93 -5.13 -9.11-4.01 -5.03 -9.17-4.01 -5.98 -10.2

Psedopotential Valence only All electron

C

XVXC



Results

Si: Matrix elements

-6

-3.5

-1

1.5

4

6.5

9

-4.86

7.82

2.14

-5.04

7.94

2.26

-5.06

8.97

3.34

Psedopotential

Valence only

All electron

C

X VXC

Remarks

ΣC : PP ≈ Val ≈ AE

ΣX : PP ≈ Val 6= AE

VXC : PP ≈ Val 6= AE

Core-valence lin.

ΣC : Small effect

ΣX : Large effect

VXC : Large effect



Results

Si: Matrix elements

-6

-1

4

9

-4.86

5.68

-5.04

5.68

-5.06

5.63Psedopotential

Valence only

All electron

C

XV

XC

Remarks




ΣX − VXC : PP ≈ Val ≈AE

Core-valence lin.

ΣC : Small effect

ΣX : Large effect

VXC : Large effect

ΣX − VXC :Small effect



Results

Si gap: G0W0 correction

0

0.2

0.4

0.6

0.8

0.7

0.59

0.55

Psedopotential

Valence only

All electronRemarks

∆Eg : PP > Val > AE

Core-valence lin.

∆Eg : Small effect.


Core-valence interaction GaAs

Results

GaAs: Matrix elements

Highest occ. state at Γ

-17.5

-15

-12.5

-10

-7.5

-5

-2.5

0

2.51.25

-12.86 -11.25

1.16

-12.5 -11.43

1.65

-17.17 -15.67

Psedopotential

Valence only

All electron

C

XVXC

Lowest unocc. state at Γ

-17.5

-15

-12.5

-10

-7.5

-5

-2.5

0 -3.55 -6.53 -10.34-3.21 -6.71 -10.67-3.58 -11.77 -16.7

Psedopotential

Valence only

All electron

C

XVXC



Results


-7.5

-5

-2.5

0

2.5

5

7.5

-4.8

6.23

0.91

-4.37

5.79

0.76

-5.23

6

-1.03

Psedopotential

Valence only

All electron

C

X VXC

Remarks




Core-valence lin.

ΣC : Small effect

ΣX : Large effect

VXC : Large effect



Results


-7.5

-5

-2.5

0

2.5

5

7.5

-4.8

5.32

-4.37

5.03

-5.23

6.43

Psedopotential

Valence only

All electron

C

XV

XC

Remarks




ΣX − VXC : PP ≈ Val 6=AE

Core-valence lin.

ΣC : Small effect

ΣX : Large effect

VXC : Large effect

ΣX − VXC :Large effect



Results

GaAs gap: G0W0 correction

0

0.25

0.5

0.75

1

1.25

0.630.66

1.2

Psedopotential

Valence only

All electron

Remarks

∆Eg : PP < Val < AE

Core-valence lin.

∆Eg : Large effect.Reduces the correction!!



Conclusions

C Si BN AlP GaAs LiF NaCl CaSe-0.6

-0.4

-0.2

0

0.2

0.4

0.6core-valence partitioning

pseudoization

Core-valence partitioning:

Strong changes in ΣX and VXC

Small changes in ΣC

∆Eg : Small changes in Si. Large in GaAsDoes NOT systematically increase the G0W0-correction.

“Pseudoization” also plays an important role.


Core-valence interaction Semicore States

Semicore states

Semicore states binding energies

MgO 6p-5p1s transition energy

TotalW L Λ Γ ∆ X ZW K-50

-40

-30

-20

-10

EF

10

Ene

rgy

[eV

]

sMg pMg s0 pO

LDA

:= 4

3.55

eV

GW

:= 5

2.41

eV

Excitation energy [eV]

ROHF(∆SCF)1 54.6CASTP21 53.8LDA2 43.5G0W

20 52.4

Experiment3 53.4

1.- C. Sousa et al. Phys. Rev. B 62,10013 (2000).2.- This work.3.- W. L. OBrien et al. Phys. Rev. B

44, 1013 (1991).


f-electron systems

f-electron systems

Motivation

Two strongly interrelated subsystems

itinerant spd statesstrongly localized f states

Intriguing physics

Heavy fermionKondo effectEtc...

Challenge to first-principle calculations:

LDA/GGA good for itinerant electronsFails for f -electrons


f-electron systems

(no)f-electron systems

CeO2

-6 -4 -2 0 2 4 6 8 10 12Energy [eV]

0

5

10

15

DO

SLDALDA-G0W0

XPS+BISXPS+XAS

E. Wuilloud et al. Phys. Rev. Lett 53, 202 (1984)

D. R. Mullins et al. Surf. Sci. 409, 307 (1998)


f-electron systems

(no)f-electron systems

Bandgaps

ZrO2 HfO2 CeO2 p-f

CeO2 p-d

ThO2

0

1

2

3

4

5

6

7E

g [eV

]LDAG0W0Expt.


f-electron systems

f-electron systems

Ce2O3

-10 -8 -6 -4 -2 0 2 4 6 8 10Energy [eV]

0

5

10D

OS

Expt.LDA

XPS XAS



f-electron systems

f-electron systems

Ce2O3

-10 -8 -6 -4 -2 0 2 4 6 8 10Energy [eV]

0

5

10D

OS

Expt.LDALDA+U

XPS XAS



f-electron systems

f-electron systems

Ce2O3

-10 -8 -6 -4 -2 0 2 4 6 8 10Energy [eV]

0

5

10D

OS

Expt.LDALDA+ULDA+U+G0W0

XPS XAS



f-electron systems

f-electron systems

Bandgaps

La2O3 Ce2O3 Pr2O3 Nd2O30

1

2

3

4

5

6E

g [e

V]

Expt.LDA+UG0W0


Conclusions

Ongoing work: Spin polarized metals

Test: Ni

W L Γ X ZW K

-10

-5

EF

5E

nerg

y [e

V]

W L Γ X ZW K

UP DOWN


Conclusions

Conclusions

GW@Wien2k

Reliable results.

Wide range of materials.

sp semiconductorsf -electron systems with empty or full f -shellsmetalsspin polarized materials

LDA+U+G0W0


Conclusions

Conclusions

Most important


Conclusions

Conclusions

Most important

Now we are enjoying it!!!


Conclusions

Last but not least

Future plans

Anisotropy of εmacro

Half metals

Efficiency Improvement

COHSEX@Wien2k + GW

BSE

QPscGW


Conclusions

Acknowledgements

Xinzheng Li (FHI, Berlin): Code development, LTM library.

Dr. Hong Jiang (FHI, Berlin): Code improvement, Spin polarization,LDA+U.

Prof. Claudia Ambrosch-Draxl (MUL; Austria): Wien2k interface andmore...

Christian Meisenbichler (MUL; Austria): MPI Paralelization

Patrick Rinke and Christoph Freysoldt (FHI, Berlin):Pseudopotentials, etc..

The boss

Matthias Scheffler


the all-electron gw method based on wien2k: implementation and applications

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