Theoretical Spectroscopy · Theoretical Spectroscopy Patrick Rinke Fritz-Haber-Institut der...

Theoretical Spectroscopy

Patrick Rinke

Fritz-Haber-Institut der Max-Planck-Gesellschaft, Berlin - Germany

Materials Theory Lecture

Patrick Rinke (FHI) Theoretical Spectroscopy TU Berlin 2012 1

Excited States in (Material) Science – Are Ubiquitous

Theoretical SpectroscopyExperiment/Spectroscopy

appropriate?

accurate?

Material Properties + Applications

appropriate?

accurate?


Excited States in (Material) Science – Are Ubiquitous

Theoretical SpectroscopyExperiment/Spectroscopy

appropriate?

accurate?

Material Properties + Applications

appropriate?

accurate?

photoemission DFT+G W0 0

from

first

principles


Band structures: photo-electron spectroscopy

- -

-

-

+

Photoemission

hn

GW

- -

-

Inverse Photoemission

-

-

hn

-

GW

- -

-+

Absorption

-

hn

BSETDDFT


Experimental: angle-resolved photoemission spectroscopy

photoemission: photon in – electron out


Experimental: angle-resolved photoemission spectroscopy

ARPES

=+1qo

Binding Energy (eV)8 6 4 2 0

Masaki Kobayashi, PhD dissertation


ARPES vs GW : the quasiparticle concept

Quasiparticle:

single-particle like excitation

A ( )k

e spectralfunction

e


ARPES vs GW : the quasiparticle concept

Quasiparticle:

single-particle like excitation

Ak(ǫ) = ImGk(ǫ) ≈Zk

ǫ− (ǫk + iΓk)

ǫk : excitation energyΓk : lifetimeZk : renormalisation

A ( )k

e quasiparticle-peak

ek

qp

Gk

e

The aim is to calculate the Green’s function G!


Green’s function and screening

-

-

-

-

-

---

--



-

-

-

-

-

-

--- -

-

-

--

+

ejected

system polarizes

tt-

-- -

- -



-

-

-

-

-

-

---

-

-

-

-

-+- -

-

-

--

+

ejected

system polarizes

tt tt

polarization is

time-dependent

hole is screened dynamically

-

--

--

- -

- -

W (r, r′, t) =

∫

dr′′ε−1(r, r′′, t)

|r′′ − r′|



-

-

-

-

-

-

---

-

-

-

-

-+- -

-

-

--

+

ejected

system polarizes

tt tt

polarization is

time-dependent

hole is screened dynamically

-

--

--

- -

- -

quasiparticle

W (r, r′, t) =

∫

dr′′ε−1(r, r′′, t)

|r′′ − r′|


GW Approximation - Screened Electrons

-

-

-

-

-

-+

-

--

G: propagator

W

GS=GW :

W: screened

Coulomb

Self-Energy

ΣGW (r, r′, t) = iG(r, r′, t)W (r, r′, t)


G0W0 Approximation in Practise

1 start from Kohn-Sham calculation for ǫKSs and φKS

s (r)

2 KS Green’s function:

G0(r, r′; ǫ) = lim

η→0+

∑

s

φKSs (r)φKS∗

s (r′)

ǫ− (ǫKSs + iη sgn(Ef − ǫKS

s ))

3 GW : χ0 = iG0G0 → W0 = v/(1− χ0v) → Σ0 = iG0W0

4 G0W0: ǫqps = ǫKSs + 〈s| Σ(ǫqps ) |s〉 − 〈s|vxc|s〉


GW Approximation - Screened Electrons

-

-

-

-

-

-+

-

--

G: propagator

W

GS=GW :

W: screened

Coulomb

Self-Energy: Σ = Σx + Σc

Σx = iGv:◮ exact (Hartree-Fock) exchange

Σc = iG(W − v):◮ correlation (screening due to other electrons)


On the importance of screening – Silicon

ǫqpnk = ǫLDAnk + 〈φnk|Σx +Σc(ǫ

qpnk)− vxc|φnk〉

exp: 1.17 eV

Hartree-Fock (exchange-only) gap much too large

Screening is very important in solids!


ARPES – GW: wurtzite ZnO

H L A G K M G

Ener

gy r

elat

ive

to(e

V)

EF

0

-2

-4

-6

ZnO G W0 0

@OEPx(cLDA)

Yan, Rinke, Winkelnkemper, Qteish, Bimberg, Scheffler, Van de Walle,

Semicond. Sci. Technol. 26, 014037 (2011)


G0W0 Band Gaps

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0Experimental Band Gap [eV]

-0.5

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

The

oret

ical

Ban

d G

ap [

eV]

LDAOEPx(cLDA)OEPx(cLDA) + G

0W

0

CdS

zb-GaNZnO

ZnS

CdSe

ZnSe

InNGe

wz-GaN

Rinke et al. New J. Phys. 7, 126 (2005), phys. stat. sol. (b) 245, 929 (2008)


Nitrides (Al,In,Ga)N – Areas of (potential) Application

LEDs and Laser Diodes

High Frequency, High Power(e.g. HEMT)

Photovoltaics

Water Splitting


Group-III nitrides

(potential) applications

solid state lighting

photovoltaics

RGB projectors

until ∼2006

debate over gap of InN


Band gap of InN

1016

1017

1018

1019

1020

1021

carrier concentration (cm-3

)

0.40.60.81.01.21.41.61.82.02.22.4

Ega

p+E

F (eV

)

Wu et alHaddad et alIoffe InstituteBriot et alTansley and FoleyRF sputteringDC sputteringButcher et alSugita et al

Figure adapted from Butcher and TansleySuperlattices Microstruct. 38, 1 (2005)


How can first principles calculations contribute?

Proposed reasons for band gap variation e.g. Butcher and Tansley Superlattices Microstruct. 38 (2005)

high carrier concentration ⇒ Moss-Burnstein effect

impurities, point defects, trapping centers

non-stoichiometry

formation of oxides and oxynitrides

metal inclusions, formation of metal clusters

First principles approach:

Density-functional theory (DFT) (ground state)

◮ atomistic control◮ stoichiometric, defect and impurity free structures

many-body perturbation theory in the GW approximation (excited state)

◮ method of choice for calculation of band gaps in solids


InN – GW band structure and Moss-Burstein effect

M/8|k

[110]|

0.0

0.5

1.0

1.5E

nerg

y [e

V]

Γ A/4|k

[001]|

w-InN

1020

1019

1018

1017


InN – GW and Moss-Burstein effect

1016

1017

1018

1019

1020

1021

carrier concentration (cm-3

)

0.60.81.01.21.41.61.82.02.2

Eg(n

) (e

V)

parabolic (meff

=0.07 m0)

G0W

0@OEPx(cLDA)

P. Rinke et al. APL 89, 161919 (2006)


Group-III nitrides

(potential) applications

solid state lighting

photovoltaics

RGB projectors

now

debate over gap of InNsettled


Future Technologies – Organic Electronics


Hybrid inorganic/organic systems (HIOS)

ZnO/PTCDA organic electronics

organic field effect transistors (OFET)

organic light emitting diodes (OLED)

organic photovoltaic cells (OPVC)

. . .

Hybrid inorganic/organic interfacesare omnipresent!



ZnO/PTCDA

ZnO/p-sexiphenyl(courtesy of S. Blumstengel)

organic electronics

organic field effect transistors (OFET)

organic light emitting diodes (OLED)

organic photovoltaic cells (OPVC)

. . .

Hybrid inorganic/organic interfacesare omnipresent!

hybrid electronics



ZnO/p-sexiphenyl(courtesy of S. Blumstengel)

Combine the best of two worlds...

inorganic materials:

stable crystal structures

good growth control

high charge carrier mobilities

organic materials:

strong light-matter coupling

large chemical space gives diverserange of properties

... or more!

great potential for:

synergies between different materials

new physics at interface


Potential for new physics at HIOS

solid organic

HIOS

Potential for new interface morphologies.


Open questions at HIOS

-

polaronic-

band-like?

What is the nature of charge carriers?

New quasiparticles?



valence band

conduction band

molecular states

-

HOMO

LUMO

-free carrier

What does the notion of “state” imply?



valence band

conduction band

molecular states

-

HOMO

LUMO

- polaron

??

-

-

polaron?

free carrier?

+ polaron ??


How do “states” line up energetically?



valence band

conduction band

molecular states

-

HOMO

LUMO

- polaron

??

-

-

polaron?

free carrier?

hybrid state?

+ polaron ??


How do “states” line up energetically?

Do hybrid “states” form?


Molecules on surfaces – a challenge for first principles

The objective is:

to develop an atomistic understanding of (nano)materials

!"#"$%&'()*$$

$$$$$$$$$$$$$+,$$

$$$$$$$$$$$$$$$$-,.$



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State-of-the-art for interfaces: density-functional theory (DFT)

local-density approximation (LDA)

generalized gradient approximation, e.g. PBE functional

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Beyond state-of-the-art:

advanced density functionals e.g. hybrid functionals or RPA

Green’s function methods e.g. GW

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Level alignment at HIOS

conductionband

valenceband

molecularstates

EF

EIB: lectrone i bnjection arrier

HIB: oleh i bnjection arrier

injection limited current:

j ∝ AT 2 exp

(

−charge injection barrier

kBT

)


Molecular levels at a surface

gas phasesurface



gas phasesurface

+

electronaffinity (EA)

ionizationpotential (IP)



gas phasesurface

+

EA

IP

-

-

-

-

image effect



gas phasesurface

+

EA

IP

-

-

-

-

image potentials

classical image potential:

metal: −1

4zsemiconductor/insulator: −

ε− 1

4(ε+ 1)

1

zε: dielectric constant



gas phaseadsorbed

+

EA

IP

-

-

-

-

renormalization

+

classical image potential:

metal: −1

4zsemiconductor/insulator: −

ε− 1

4(ε+ 1)

1

zε: dielectric constant


The screened Coulomb interaction W

Work with screened coulomb interaction:

W (r, r′, t) =

∫

dr′′ε−1(r, r′′, t)

|r′′ − r′|

ε(r, r′′, t): dielectric function

The challenge is to calculate W (r, r′′, t) from first principles!


The screened Coulomb interaction W

Work with screened coulomb interaction:

W (r, r′, t) =

∫

dr′′ε−1(r, r′′, t)

|r′′ − r′|

ε(r, r′′, t): dielectric function

The challenge is to calculate W (r, r′′, t) from first principles!

we use random-phase approximation (RPA) for W :

= + + + ...

v: bare Coulomb interaction

c0: Kohn-Sham polarizability

W

formally scales with (system size)4


From screening to quasiparticle energies

-

-

-

-

-

-+

-

--

G: propagator

W

GS=GW :

W: screened

Coulomb

Quasiparticle energies: G0W0 scheme

electron addition and removal energies

correction to DFT eigenvalues ǫDFT

nk :

ǫqpnk = ǫDFT

nk +ΣG0W0

nk (ǫqpnk)− vxcnk

includes image effect


GW for surfaces – CO on NaCl

Na ClOC

HOMO-LUMO gap of CO:

gap/eV LDA G0W0 Exp.@LDA

free CO : 6.9 15.1 15.8∗

CO@NaCl: 7.4 13.1∗Constants of Diatomic Molecules (1979),

Phys. Rev. Lett. 22, 1034 (1969)

surface polarization significant here

C. Freysoldt, P. Rinke, and M. Scheffler, Phys. Rev. Lett. 103, 056803 (2009)


GW for surfaces – CO on Ultrathin NaCl on Ge

CO

NaCl

Ge

Supported ultrathin films are novel nano-systems withunexpected features (PRL 99, 086101 (2007))

Additional electrons/holes on CO see two interfaces

Will the CO gap depend on NaCl thickness?


GW for surfaces – CO on Ultrathin NaCl on Ge

12.512.612.712.8

G0W

0@LDA

2 3 4NaCl layers

7.47.57.67.77.8

DFT-LDA

5σ-2

π∗ spl

ittin

g (e

V)

3.03.13.23.33.4

2π∗

2 3 4NaCl layers

-2.1-2.0-1.9-1.8-1.7

5σ

G0W

0 cor

rect

ions

(eV

)Polarization at NaCl/Ge interface gives layer-dependent CO gap

⇒ molecular levels can be tuned by polarization engineering:◮ interesting for molecular electronics and quantum transport

C. Freysoldt, P. Rinke, and M. Scheffler, PRL 103, 056803 (2009)



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Electron gas at TTF/TCNQ interface

TTF and TCNQ crystals have

large band gap

but 2 dimensional electron gasobserved at interface

Nat. Mat. 7, 574 (2008)

What is the origin?

TTF

TCNQ


Electron gas at TTF/TCNQ interface

-3 -2 -1 0 1 2 3E-E

F [eV]

0

20

40

DO

S

DFT-PBE density of states

DFT-PBE:

predicts metallic interface

0.12 e/molecule charge transferto TCNQ

in seeming agreement withexperiment

TTF

TCNQ


TTF/TCNQ dimer at infinite separation

TTF TCNQ

EAIP

exp PBE expPBE

4.0

6.7

9.5

7.0

5.6

2.81.0

!

IP(TTF) > EA(TCNQ)

erroneous charge transfer


Ionisation Potential, Electron Affinity and (Band) Gaps

Could use total energy method to compute

ǫs = E(N ± 1, s)− E(N)

Ionisation potential: minimal energy to remove an electron

I = E(N − 1)− E(N)

Electron affinity: minimal energy to add an electron

A = E(N)− E(N + 1)

(Band) gap: Egap = I −A


∆SCF versus eigenvalues for finite systems

∞≈≈

data courtesy of Max Pinheiro


Ionisation Potential, Electron Affinity and Band Gap

so what about ∆SCF for excitations?

∆SCF: ǫs = EDFT(N)− EDFT(N ± 1, s)

reasonably good for I and A of small finite systems

but:

most functionals suffer from delocalization or self-interaction error(Science 321, 792 (2008))

⇒ the more delocalized the state, the larger the error

only truely justified for differences of ground states◮ ionisation potential, electron affinity◮ excited states that are ground states of particular symmetry

difficult to find excited state wavefunction (density)

excited state density is not unique

separate calculation for every excitation needed◮ not practical for large systems or solids



culprit: self-interaction error in PBE

add Hartree-Fock (HF) exchange to removeself-interaction

⇒ PBE0-like hybrid functional:

Exc(α) = α(EHFx − EPBE

x ) + EPBEc



0 0.2 0.4 0.6 0.8 1 α

-2

0

2

4

∆[e

V]

Experiment

PBE0(α)

artificial charge transfer

PBE0-like hybrid functional:

Exc(α) = α(EHFx − EPBE

x ) + EPBEc



0 0.2 0.4 0.6 0.8 1 α

-2

0

2

4

∆[e

V]

Experiment

PBE0(α)

artificial charge transfer

How to choose α?


Performance of G0W0

TTF TCNQ

S

CH

H

NC

EA

IP

expG0W0 exp G0W0

6.7 6.7

9.5 9.7

3.7

-0.7

2.8

C6H4S4

all values in eV

C12H4N4

4.6

Exp.: JACS 112, 3302 (1990), J. Chem. Phys. 60, 1177 (1974),Adv. Mat. 21,1450 (2009),Mol. Cryst. Liquid Cryst. Inc. Nonlin. Opt. 171, 255 (1989)


TTF/TCNQ dimer – implications for interface

no charge transfer, but small charge rearrangement

2D electron gas not due to charge transfer

HOMO

LUMO

PBE optimal !

electron density difference

optimal-α calculations for interface in progress



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Acknowledgements

GW for surfaces

Christoph Freysoldt

Nitrides

Abdallah QteishMomme WinkelnkemperJorg Neugebauer

GW in FHI-aims

Xinguo Ren

Support and Ideas

Matthias Scheffler

TTF-TCNQ

Viktor AtallaMina Yoon

gwst code

Rex Goby

FHI-aims code

Volker Blumthe whole FHI-aims developer team


Theoretical Spectroscopy · Theoretical Spectroscopy Patrick Rinke Fritz-Haber-Institut der...

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