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

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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

principles

Band structures: photo-electron spectroscopy

Photoemission

Inverse Photoemission

Absorption

BSETDDFT

Experimental: angle-resolved photoemission spectroscopy

photoemission: photon in – electron out

Binding Energy (eV)8 6 4 2 0

Masaki Kobayashi, PhD dissertation

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

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

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

Green’s function and screening

ejected

system polarizes

ejected

system polarizes

polarization is

time-dependent

hole is screened dynamically

W (r, r′, t) =

dr′′ε−1(r, r′′, t)

|r′′ − r′|

ejected

system polarizes

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

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

2 KS Green’s function:

G0(r, r′; ǫ) = lim

η→0+

φKSs (r)φKS∗

s (r′)

ǫ− (ǫKSs + iη sgn(Ef − ǫKS

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

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

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]

LDAOEPx(cLDA)OEPx(cLDA) + G

zb-GaNZnO

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

carrier concentration (cm-3

0.40.60.81.01.21.41.61.82.02.22.4

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

[110]|

Γ A/4|k

[001]|

InN – GW and Moss-Burstein effect

carrier concentration (cm-3

0.60.81.01.21.41.61.82.02.2

parabolic (meff

=0.07 m0)

0@OEPx(cLDA)

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

Group-III nitrides

(potential) applications

solid state lighting

photovoltaics

RGB projectors

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

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

-free carrier

What does the notion of “state” imply?

valence band

conduction band

molecular states

- polaron

polaron?

free carrier?

+ polaron ??

How do “states” line up energetically?

valence band

conduction band

molecular states

- 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

EIB: lectrone i bnjection arrier

HIB: oleh i bnjection arrier

injection limited current:

j ∝ AT 2 exp

−charge injection barrier

Molecular levels at a surface

gas phasesurface

electronaffinity (EA)

ionizationpotential (IP)

gas phasesurface

image effect

gas phasesurface

image potentials

classical image potential:

metal: −1

4zsemiconductor/insulator: −

ε− 1

4(ε+ 1)

zε: dielectric constant

gas phaseadsorbed

renormalization

classical image potential:

metal: −1

4zsemiconductor/insulator: −

ε− 1

4(ε+ 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

formally scales with (system size)4

From screening to quasiparticle energies

G: propagator

GS=GW :

W: screened

Coulomb

Quasiparticle energies: G0W0 scheme

electron addition and removal energies

correction to DFT eigenvalues ǫDFT

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

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

2 3 4NaCl layers

7.47.57.67.77.8

DFT-LDA

π∗ spl

3.03.13.23.33.4

2π∗

2 3 4NaCl layers

-2.1-2.0-1.9-1.8-1.7

)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?

Electron gas at TTF/TCNQ interface

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

F [eV]

DFT-PBE density of states

DFT-PBE:

predicts metallic interface

0.12 e/molecule charge transferto TCNQ

in seeming agreement withexperiment

TTF/TCNQ dimer at infinite separation

TTF TCNQ

exp PBE expPBE

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

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

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

Experiment

PBE0(α)

artificial charge transfer

How to choose α?

Performance of G0W0

TTF TCNQ

expG0W0 exp G0W0

6.7 6.7

9.5 9.7

C6H4S4

all values in eV

C12H4N4

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

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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