GW and Bethe-Salpeter Equation Approach to Spectroscopic...
Transcript of GW and Bethe-Salpeter Equation Approach to Spectroscopic...
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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
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First-Principles Study of Material Properties
+
= iGW
Fermi sea
Fermi sea
(excitonic)
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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
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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!
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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?
…
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Diagrammatic Expansion of the Self Energy in Screened Coulomb Interaction
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Hybertsen and Louie (1985)
H = Ho + (H - Ho)
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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
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Quasiparticle Band Structure of Germanium
Theory: Hybertsen & Louie (1986)
Photoemission: Wachs, et al (1985)
Inverse Photoemission:Himpsel, et al (1992)
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Self-energy Corrections in Graphene Nanoribbons
-states
NFE- sheet states
-states
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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
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Rohlfing & LouiePRL,1998.
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Optical Absorption Spectrum of SiO2
Chang, Rohlfing& Louie.PRL, 2000.
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M. Rohfling and S. G. Louie, PRL (1998)
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Both terms important!
repulsive
attractive
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Rohlfing & LouiePRL, 1998.
Optical Absorption Specturm of GaAs
Bound excitons
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Optical Absorption Spectrum of SiO2
Chang, Rohlfing& Louie.PRL, 2000.
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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!
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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
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• 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
+
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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)
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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)
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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
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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)
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(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)
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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)
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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)
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(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.
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Experimental Absorption Spectrum ofSingle Suspended (21,21) Metallic SWCNT
F. Wang, et al (2007)
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Exciton in (21,21) Metallic SWNCT: Theory vs. Experiment
Free electron-hole interbandtransition picture
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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)]
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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
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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.)
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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)
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Graphene Electronic Structure
kx
ky
Ener
gy
kx' ky'
E
unoccupied
occupied
E =hvF
r k
EF
E2 = p2c2
2D massless Dirac fermion system
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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)
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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)
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Forces in the Photo-Excited State:Self-trapped Exciton
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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)
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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).
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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
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SiO2 ( -quartz): optical properties
• Oxygen• Silicon
[1] Ismail-Beigi & Louie (2004)[2] Philipp, Sol. State. Comm. 4 (1966)
[1]
Emission at ~ 3 eV!
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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)
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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
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Atomic rearrangement for STE
No activation barrier!
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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
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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
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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).
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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
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Rohlfing & Louie,PRL, 1998.
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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
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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)
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• 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)
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LDA+U as a starting mean-field H for GW quasiparticle calculations
- bcc hydrogen - ZnS
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Hybertsen and Louie (1985)
H = Ho + (H - Ho)
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GW gap
GGA K-S gap
EQP
bcc Hydrogen
Energy gap at rs = 4
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bcc Hydrogen
Energy gap at rs = 4
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bcc Hydrogen
Energy gap at rs = 4
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bcc Hydrogenenergy gap vs. rs
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17.2=s
r
Expt:12.8eV
(rsVQMC = 2.1 ± 0.1)
bcc Hydrogenenergy gap vs. rs
Kioupakis, Zhang, & Louie (2006)
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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
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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.
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Collaborators
Mark HybertsenEric ShirleyJohn NorthrupMichael RohlfingEric ChangSohrab Ismail-BeigiCatalin SpataruJack DeslippeDavid Prendergast
Li YangMei-Yin ChouYoung-Woo SonMarvin CohenJeff NeatonManos KioupakisPeihong ZhangTakashi Miayake…