GW and Bethe-Salpeter Equation Approach to Spectroscopic ...nano-bio.ehu.es/.../GW_and_Bethe-Salpeter...Louie.pdf · Steven G. Louie Department of Physics, University of California

GW and Bethe-Salpeter Equation Approach toSpectroscopic Properties

Steven G. Louie

Department of Physics, University of California at Berkeleyand

Materials Sciences Division, Lawrence Berkeley National Laboratory

Supported by: National Science FoundationU.S. Department of Energy

First-Principles Study of Material Properties

+

= iGW

Fermi sea

Fermi sea

(excitonic)

Content

• Quasiparticle excitations

- The GW approximation- Applications to solids, surfaces and nanostructures

• Excitons, optical response, and forces in the excited state

- The Bethe-Salpeter Equation- Applications to crystals, surfaces, nanotubes, self-

trapped excitons

• Some more-correlated systems

Quasiparticle Excitations

Kohn-Sham Eigenvalues QP Energies

One simple example: the Homogeneous Interacting Electron System

Standard K-S equation:

1

2

2+Vext +VH +

Exc(r)

(r)=

KS(r)

Vext +VH = constant

Vxc (r) =Exc(r)

constant Free electron dispersion (m* = me, infinite

lifetime, etc.)

WRONG!

Additional Theoretical Issues

• Kohn-Sham formulation is only one approach to DFT.- not unique- different formulation different eigenvalues

• How shall we interpret the K-S eigenvalues?- electron addition energies?- optical transition energies?

…

Diagrammatic Expansion of the Self Energy in Screened Coulomb Interaction

Hybertsen and Louie (1985)

H = Ho + (H - Ho)

Quasiparticle Band Gaps: GW results vs experimental values

Compiled byE. Shirley andS. G. Louie

Materials included:

InSb, InAsGe GaSbSiInPGaAsCdSAlSb, AlAsCdSe, CdTeBPSiCC60GaPAlPZnTe, ZnSec-GaN, w-GaNInSw-BN, c-BNdiamondw-AlNLiClFluoriteLiF

Quasiparticle Band Structure of Germanium

Theory: Hybertsen & Louie (1986)

Photoemission: Wachs, et al (1985)

Inverse Photoemission:Himpsel, et al (1992)

Self-energy Corrections in Graphene Nanoribbons

-states

NFE- sheet states

-states

Si(111) 2x1 Surface

Measured values: Bulk-state qp gap 1.2 eV Surface-state qp gap 0.7 eV Surface-state opt. gap 0.4 eV

Rohlfing & LouiePRL,1998.

Optical Absorption Spectrum of SiO2

Chang, Rohlfing& Louie.PRL, 2000.

M. Rohfling and S. G. Louie, PRL (1998)

Both terms important!

repulsive

attractive

Rohlfing & LouiePRL, 1998.

Optical Absorption Specturm of GaAs

Bound excitons

Optical Absorption Spectrum of SiO2

Chang, Rohlfing& Louie.PRL, 2000.

Nanostructures

• Size and restricted geometry => quantum confinement,enhanced many-electron interaction, reduceddimensionality, and symmetry effects

– Novel properties and phenomena– Useful in applications

Size

Bawendi Group: Colloidal CdSequantum dots dispersed in hexane.

• Small can be different!

Optical Excitations in Carbon Nanotubes

• Recent advances allowed the measurement of optical response of wellcharacterized, individual SWCNTs.

• Response is quite unusual and cannot be explained by conventionalpictures.

• Many-electron interaction (self-energy and excitonic) effects are veryimportant => interesting physics

(n,m) carbon nanotube

• Many-electron interaction effects

- Quasiparticles and the GW approximation

- Excitonic effects and the Bethe-Salpeterequation

• Single-walled carbon nanotubes

- Absorption spectra

- Exciton binding energies and wavefunctions

- Radiative lifetime, …

First-principles Study of Optical Properties

+

Quasiparticle Self-Energy Corrections

• Metallic tubes -- stretch of bands by ~15-25% (velocity renormalization)• Semiconductor tubes -- large opening (~ 1eV) of the gap

(8,0) semiconducting SWCNT(10,10) metallic SWCNT

Spataru, Ismail-Beigi, Benedict & Louie, PRL (2004)

GW Quasiparticle Band Dispersion of Metallic CNTs

Quasiparticle energy corrections:

• larger compared to graphite• increase with increasing diameter

Quasiparticle Fermi Velocities (106 m/s)

LDA QP GW shift

(3,3) 0.56 0.65 15%

(5,5) 0.72 0.85 19%

(10,10) 0.81 1.00 24%

Graphene 0.82 1.04 28%

Eq

p(e

V)

ELDA-EF

(10,10)

Absorption Spectrum of Semiconducting (8,0) Carbon Nanotube

• Long-range attractive electron-hole interaction• Spectrum dominated by bona fide and resonant excitons• Large binding energies ~ 1eV! [Experimental verification: Wang, Heinz et al, (2005); Ma, Fleming, et al.

(2005); Maultzsch, Molinari, et al, (2005), Avouris, et al …]

Spataru, Ismail-Beigi, Benedict &Louie, PRL 92, 077402 (2004)

(Not Frenkel-like)

| (re,rh)|2

1.41b1.441.67b1.741.19b1.21(11,0)

2.16a2.392.31a2.391.07a1.00(10,0)

1.17a1.161.88a1.801.60a1.55(8,0)

2.43a2.503.14a3.001.29a1.20(7,0)

Exp.TheoryExp.TheoryExp.Theory

E22/E112nd transition

(E22)1st transition

(E11)

Optical transition energies (in eV) of four semiconducting CNTs

aS. Bachilo, et al. (2002), bY-Z Ma, et al, (2005)

• Important Physical Effects: - band structure (~ eV shift each) - quasiparticle self-energy

- excitonic• Transport gap optical gap

Spataru, Ismail-Beigi, Benedict & Louie, PRL (2004)

(7,0) (8,0)

(10,0) (11,0)

Optical Spectrum of Semiconducting Carbon SWNTs

• Excitonic effects are equally dominant in BN nanotubes and Si nanowires!

Spataru, Ismail-Beigi, Benedict & Louie (2004)

Absorption Spectrum of (3,3) Metallic Carbon Nanotube

• Existence of bound excitons in metal tubes! (Eb = 86 meV)• Due to ineffective screening in 1D and symmetric gap• Similar results for the (10,10) and larger metallic tubes

EF

Spataru, Ismail-Beigi, Benedict & Louie,PRL (2004)

Optical Absorption Spectra of Two Metallic SWCNTs

(10,10)

(12,0)

Excitons in Metallic CNTs

• One bright exciton pervan Hove singularity

• Exciton binding energyEb ~ 50-100 meV

• Eb weakly dependent ontube diameter

70 meV

60 meV

(5,5)

120 meV60 meVEb22

50 meV50 meVEb11

(12,0)(10,10)

Exciton binding enegies

Deslippe, Spataru, Prendergast & Louie,Nanoletters (2007)

(10,10) Metallic SWCNT Peak Shape Comparison(broadened with linewidth of 80 meV)

Interband transitions model

With bound exciton (present theory)

• Significant optical line-shape difference should be observable

Energy (eV)

Deslippe, Spataru, Prendergast & Louie,Nanoletters (2007)

Note: black curveis shifted by 50meV to align withred curve.

Experimental Absorption Spectrum ofSingle Suspended (21,21) Metallic SWCNT

F. Wang, et al (2007)

Exciton in (21,21) Metallic SWNCT: Theory vs. Experiment

Free electron-hole interbandtransition picture

Exciton in (21,21) Metallic SWNCT: Theory vs. Experiment

Exciton Theory E

b = 50 meV

Rex

= 3.1 nm

Theory

Experiment

(Note: 80 meV broadening used in theory)

Wang, et al, to be pubished (2007)

[Additional evidence seen in field-enhanced photocurrent measurements,Mohite, et al (2007)]

Science 299,1874 (2003)

7 nm

5 nm

3 nm

2.5 nm

2 nm

1.3 nm

Hydrogen terminated Si nanowires

STM measurement of SiNW on graphite

Optical absorption in Si wireA

bsor

ptio

nOptical Spectrum of d=1.2 nm Si Nanowire

Exciton binding energy > 1 eV!Yang, Spataru, Louie & Chou (2006)

= 3.2 eV(~3.4 eV expt.)

Graphene Nanoribbons

• Phenomenon of electric field-induced half-metallicity

– Tunable spin carriers of one type (100%spin polarization)

– Could be useful for nanoscale spingeneration and injection

• Optical response is also dominated by excitons

Son, Cohen and Louie, Nature (2006)Son, Cohen and Louie, PRL (2006)Yang, Son, Cohen and Louie, (2007)

Graphene Electronic Structure

kx

ky

Ener

gy

kx' ky'

E

unoccupied

occupied

E =hvF

r k

EF

E2 = p2c2

2D massless Dirac fermion system

Graphene Nanoribbons with Homogenous Edges & Passivated -bonds

Armchair Graphene Nanoribbons(N-AGNRs)

Simple tight-binding:

Metal: Na = 3p+2 Semiconductor: Na = 3p or 3p+1

Zigzag Graphene Nanoribbons(N-ZGNRs)

Simple tight-binding: Always metal

Ab initio calculations predicted all GNRs have gaps!

Son, Cohen and Louie, PRL (2006)

Quasiparticle Band Structure and Optical Spectrum of 10-AGNR

Armchair-edgenanoribbon

• Width of w ~ 1.1nm• Large exciton binding energy of Eb ~1.3 eV• Similar strong exciton effects in other

nanoribbons

Yang, Park, Cohen and Louie (2007)

Forces in the Photo-Excited State:Self-trapped Exciton

Forces in Excited State

• For many systems, photo-induced structural changesare important– differences between absorption and luminescence– self-trapped excitons– molecular/defect conformation changes– photo-induced desorption

• Need excited-state forces– structural relaxation– luminescence study– molecular dynamics, etc.

• GW+BSE approach gives accurate forces in photo-excited state

Ismail-Beigi & Louie, Phys. Rev. Lett. 90, 076401 (2003)

Excited-state Forces

ES = E0 + S

RES = RE0 + R S

E0 & RE0 : DFT

S : GW+BSE

Ismail-Beigi & Louie, Phys. Rev. Lett. 90, 076401 (2003).

Verification on molecules

Ismail-Beigi & Louie, Phys. Rev. Lett. 90, 076401 (2003).

Excited-state force methodology

• Proof of principle: tests on molecules

- CO, NH3, …

• GW-BSE force method works well

• Forces allow us to efficiently find excited-stateenergy minima

SiO2 ( -quartz): optical properties

• Oxygen• Silicon

[1] Ismail-Beigi & Louie (2004)[2] Philipp, Sol. State. Comm. 4 (1966)

[1]

Emission at ~ 3 eV!

Self-trapped exciton (STE) in SiO2 ( -quartz)

Triplet STE has 1 ms and ~ 6 eV Stokes shift [1]

[1] e.g. Itoh, Tanimura, &Itoh, J. Phys. C 21 (1988).

1. Start with 18 atom bulk cell

2. Randomly displace atoms by ±0.02 Å

3. Relax triplet exciton state

4. Repeat steps 2&3: same final config.

Ismail-Beigi & Louie, PRL (2005)

Structural Distortion from Self-Trapped Exciton in SiO2

Final configuration: Broken Si-O bond Hole on oxygen Electron on silicon Si in planar sp2 configuration

Ismail-Beigi & Louie, PRL (2005)

• Oxygen• Silicon

Atomic rearrangement for STE

No activation barrier!

Electron-Hole Wavefunction of Self-Trapped Exciton in SiO2

Hole probability distributionwith electron any where inthe crystal

Electron probabilitydistribution given thehole is in the colored box

Electron & Hole Distributions of Self-Trapped Exciton in SiO2

Final configuration: Broken Si-O bond Hole on oxygen (brown) Electron on silicon (green) Si in planar sp2 configuration

Ismail-Beigi & Louie, PRL (2005)

• Oxygen

• Silicon

Constrained DFT Calculations

Constrained LSDA: DFT with excited occupations

Problems:

• Relaxes back to ideal bulk from random initial displacements: excited-state energy surface incorrectly has a barrier.

• Large initial distortions needed for STE [1,2]

• Predicted Stokes shift and STE luminescence energy are very poor to correlate with experiments

[1] Song et al., Nucl. Instr. Meth. Phys. Res. B 166-167, 451 (2000).[2] Van Ginhoven and Jonsson, J. Chem. Phys. 118, 6582 (2003).

STE in SiO2: Comparison to Experiment

2.14

6.37

6.2-6.4

Stokes shift(eV)

----4.12CLSDA (forced)

0.48, 0.65,0.70

2.6, 2.74,2.75, 2.8

Expt. [1-6]

GW+BSE 2.85

Luminescencefreq.: T (eV)

LuminescencePol || z (*)

0.72

1. Tanimura et al., Phys. Rev. Lett. 51, 423 (1983).2. Tanimura et al., Phys. Rev. B 34, 2933 (1986).3. Itoh et al., J. Phys. C 21, 4693 (1988).4. Itoh et al., Phys. Rev. B 39, 11183 (1989).5. Joosen et al., Appl. Phys. Lett. 61, 2260 (1992).6. Kalceff & Phillips, Phys. Rev. B 52, 3122 (1996).

(*) Pol =Iz IxyIz + Ixy

Rohlfing & Louie,PRL, 1998.

Molecular energy levels at metal-organic interfaces

Metal-organic contacts Energy level diagram

Ubiquitous in nanoscale devices

Single-molecule junctions, organicelectronics, passivators fornanoparticle surfaces, etc

Fermi Energy

E

Metal

z

Affinity Level

Ionization Level

vacuum

HOMO

LUMO

• Frontier molecular orbital alignment?• HOMO-LUMO gap?• Implications for charge transport?

Physical effects

• Charge transfer (interface dipoles)• Quantum mechanical (electronic) coupling• Surface polarization

10.5 eV

Frontier Levels in the gas phase

Experiment:

IP – EA = 10.4 – 10.6 eV

5.1 eV

LDA GW • LDA underestimates the gapby a factor of 2

• GW HOMO-LUMO gapagrees with experiment (IP-EA)

• LUMO predicted to be abovethe vacuum level in GW, inagreement with experiment

Gas-phase benzene: HOMO-LUMO gap

Neaton, Hybertsen, Louie, PRL (2006)

• HOMO-LUMO gaps of aromaticmolecules are reduced at metal contacts

• Nonlocal electronic correlations betweenthe molecule and substrate areresponsible

Benzene @ graphite: Energy level

diagram

EF

Graphite

Metal-molecule interface

7.3 eV 10.5 eV

Isolated

molecule

Benzene @ graphite: Frontier electronic orbitals

Excited electronic states at the organic-inorganic interface

Neaton, Hybertsen, Louie, Phys. Rev. Lett. (2006)

(5.16)(5.05)

LDA+U as a starting mean-field H for GW quasiparticle calculations

- bcc hydrogen - ZnS

Hybertsen and Louie (1985)

H = Ho + (H - Ho)

GW gap

GGA K-S gap

EQP

bcc Hydrogen

Energy gap at rs = 4

bcc Hydrogen

Energy gap at rs = 4

bcc Hydrogen

Energy gap at rs = 4

bcc Hydrogenenergy gap vs. rs

17.2=s

r

Expt:12.8eV

(rsVQMC = 2.1 ± 0.1)

bcc Hydrogenenergy gap vs. rs

Kioupakis, Zhang, & Louie (2006)

Zhang, Miyake, Cohen and Louie (2006)

U = 8 eVJ = 1 eV

LDA GW(LDA)

LDA+U GW(LDA+U)

Egexpt

d states (expt)

Energy states of ZnS

Summary • The GW-BSE approach is a powerful method for

studying quasiparticle excitations and photo-excited states of condensed matter.

• Very robust for a number of moderately correlated

systems – crystals, surfaces, polymers, nanostructures… • Present methods can handle up to ~ 50-100 atoms per

supercell. • Need improvements to address larger and more correlated

systems.

Collaborators

Mark HybertsenEric ShirleyJohn NorthrupMichael RohlfingEric ChangSohrab Ismail-BeigiCatalin SpataruJack DeslippeDavid Prendergast

Li YangMei-Yin ChouYoung-Woo SonMarvin CohenJeff NeatonManos KioupakisPeihong ZhangTakashi Miayake…