Theoretical Framework for Electronic & Optical Excitations,...
Transcript of Theoretical Framework for Electronic & Optical Excitations,...
CECAM, Berlin, 8/4&6/12 1
Theoretical Framework for Electronic & Optical Excitations,
the GW & BSE Approximations and Considerations for Practical Calculations
Mark S HybertsenCenter for Functional Nanomaterials
Brookhaven National Laboratory
HoW exciting!Hands-on Workshop on Excitations in Solids 2012
CECAM, Berlin, Germany
Work supported by Brookhaven Science Associates, LLC under Contract No. DE-AC02-98CH10886 with the U.S. Department of Energy.
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Do you speak GW?
1965:� Hedin develops an approach for systematic approximations for the electron self
energy operator in many-body perturbation theory that naturally includes screening.
− Lowest order term: ΣΣΣΣ = iGW
1980’s & 1990’s:� Reliable calculations for real materials emerge & “GW” works!� Methodologies diversify & technical questions bubble …
2000’s to today: Which “GW” ?
� G0W0, GW0, G0W, GW, self consistency, vertex corrections, …
2010s:
� Efficiency: Complexity one order higher than ground state (at least)
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Resources
Books for fundamentals of many-body physics techniques and applications –� Fetter and Walecka, “Quantum Theory of Many-Particle Systems” (Dover)
− Old school: excellent formal development
� Mahan, “Many Particle Physics” (3rd edition)− Common text-book: more focused on exemplary MB problems
� Haug and Jauho, “Quantum Kinetics in Transport and Optics of Semiconductors”(2nd edition, Springer)
− Focused on non-equilibrium theory and applications
Review articles –� Hedin and Lundqvist, Solid State Physics, vol. 23, pp. 1-181, 1969
− Strong exposition of fundamentals; no optics / BSE; materials discussion dated & limited
� Aulbur, Jonsson and Wilkins, Solid State Physics, vol. 54, pp. 1-218, 2000− Reviews fundamentals; discussion of computational issues c2000;no optics / BSE; diverse materials examples
� Onida, Reining and Rubio, Rev. Mod. Phys, vol. 74, pp. 601-659, 2002− Includes both GW and BSE; includes TD-DFT; materials examples and exposition emphasize optics
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Outline for Lecture I
Introduction: Electronic Excitations
Theoretical Framework: Green’s Function Approach
Hedin’s Equations & the GW Approximation (1965)
Physical Ingredients, Practical Considerations for Real Materials& Illustrative Examples (c1990)
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Independent Electron Model
Neutral atom or molecule� Electrons sequentially fill discrete
quantum mechanical levels a la Fermi
� Prototypical electronic excitation:− Ionization energy threshold:
IP = E(N-1) −−−− E(N)
Evac
N electrons
Evac
N−−−−1 electrons
E
kEF
kx
ky
Metallic solid� Electrons sequentially fill a continuum
of Bloch wave states below the Fermi Energy
� Prototypical electronic excitations:− Thermal distribution of electrons & holes
� Fundamental to conductivity, heat capacity, …
� Also characterized by electron removal energies (photoemission spectra)
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Independent Electron Model: Empirical Pseudopotentials
Ingredients –� The low order fourier components, screened local potential: Vloc(G)
� Angular momentum resolved, atom centered potentials (non-local): Vnl(k+G,k+G’ )� Fit key transition energies (e.g. 11 parameters, including spin-orbit, for InP)
Results for semiconductors –� Full band structure & optical spectra� Good agreement w/ photoemission
� Adequate band masses
Similar approach for metals ���� Fermi surfaces
Chelikowsky & Cohen PRB, 1976
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Confronting the Many Interacting Electrons
Landau Fermi Liquid Theory� Low energy properties of the interacting
system described by quasiparticleexcitations with weak residual interactions
Emphasis on Model Hamiltonians
Quantum Monte Carlo Methods
Density Functional Theory
Hartree-Fock + Configuration Interaction Theory
� Singles, doubles, …
Coupled-cluster Theory
Many-Body Perturbation Theory
Quantum Monte Carlo Methods
Many-body Physics Ab initio Materials & Chemistry
one-body electron-electron Coulomb interaction
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Density Functional Theory
Hohenberg-Kohn-Sham� Ground state energy universal functional of electron density – variational
� Fictitious system of independent particles in an effective potential
− Today: many approximate functionals (LDA, GGA, Hybrids, …) ���� efficient theory for ground state properties
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Density Functional Theory
What about the Kohn-Sham bandstructure ?� Except for the highest occupied state,
no physical meaning!� In practice, often a good guide,
but band gaps wrong!− Reliable DFT for bulk Silicon
� Hamann, PRL, 1979
Fundamental: there is a discontinuity in δδδδExc/δδδδn
� Sham & Schluter, PRL, 1983; Perdew & Levy, PRL, 1983
− Note: Density matrix functional theory different� Recent work of Wei Tao Yang
SiliconExpt: 1.17 eVLDA: ~0.5 eVKS: 0.66 eV∆∆∆∆xc: 0.58 eV
Godby, Schluter & Sham, PRL, 1986
( ) ( )( ) ( ) ( )( ) xcKSgg NENENENEE ∆+=−−−−+= ,11 ε
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Hartree-Fock (Mean-Field) Theory
Condition on the spin-orbitals to optimize the ground-state energy:
Postulate a variational wavefunction: Slater determinant form –
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Hartree-Fock (Mean-Field) Theory
Mean-field theory with independent orbital occupation by pairs of electrons (spin ‘restricted’ Hartree-Fock)
− Adequate for accurate molecular structure in chemistry
− Poor binding energies, …
Resulting in the HF equations for the orbitals –
exchange interaction
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Hartree-Fock (Mean-Field) Theory
Koopman’s Theorem for Electronic Excitations:
� Condition: no orbital relaxation(self consistent change in {φn} for the ion)
− Low accuracy for ionization levels in molecules
− Semiconductors: Eg too large
Evac
N−−−−1 electrons
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Hartree-Fock (Mean-Field) Theory: Metals
Electron-gas Model
kx
ky
0 0.5 1 1.5-20
-15
-10
-5
0
ΣΣ ΣΣ x(k
) (e
V)
k/kF
ΣΣΣΣx
-25 -20 -15 -10 -5 0 5 100
DO
S (
arb)
E - EF (eV)
Density of States
free
HF
0 0.5 1 1.5-25
-20
-15
-10
-5
0
5
10
15
20
25
εε εε k-
EF
(eV
)
k/kF
Dispersion
freeHF
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Outline for Today
Introduction: Electronic Excitations� Electronic excitations: electronic addition or removal energies
� Particle like excitations (“Quasiparticles”) fundamental to understand solids� Correlation beyond mean-field (HF) is essential
� The KS eigenvalues are not a fundamentally sound approach
Theoretical Framework: Green’s Function Approach
Hedin’s Equations & the GW Approximation (1965)
Physical Ingredients & Practical Considerations for Real Materials& Illustrative Examples (c1990)
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Green’s Function Framework: Physical Motivation
Physical impact of electron-electron interactions:
kx
ky
kx
kyFinite
lifetimeEnergy &
momentumconservation���� ΓΓΓΓk ~ (Ek-EF)2
Quasiparticle excitation energies:
EEQP
Pro
babi
lity
Nointeractions
ΓΓΓΓQP
incoherent
Distributionfor injected electron
with interactions
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One particle Green’s function, N-electron system:� Temporal & spatial evolution of an added electron –
Single Particle Green’s Function in Many-Body Theory
� Spectral representation:
− Poles of ImG correspond to excitations
Note 2 nd quantization:ψψψψ a field operator
Lehmann representation & excitation energies:� Amplitudes from exact excited states s of N+1 / N-1 electron systems:
− Set {fs(r)} complete, but not orthonormal
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Spectral representation:� Solutions of the homogeneous (Schroedinger) equation:
� Including infinitesimal to distinguish forward/backward propagation:
� Spectral function � Density of states:
Green’s Function: Equation of Motion
One-body case:
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Define self energy operator to satisfy the one particle Green’s function:� Equation of motion –
− Self energy operator ΣΣΣΣ depends on G, v(r,r’)
− Requires approximate treatment− Generally complex & non-hermitian
MBPT: Single Particle Green’s Function
Another spectral representation:� Homogeneous solutions –
� Green’s function –
− {vk,E(r)} “left” solutions; form biorthogonal/complete set with {uk,E(r)}
− εεεεk,E complex− Set of solutions needed for every energy E
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Green’s Function: Quasiparticle Equation
EEQP
ΓΓΓΓQP
incoherent
ImGk(E)
Focus on the energy region near the quasiparticle energies:
− Evaluate ΣΣΣΣ at the quasiparticle energy
− Self energy non-hermitian ���� Ek complex
FundamentalEquation
SpectralDensity
A(E) = ππππ-1|ImG(E)|
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Outline for Today
Introduction: Electronic Excitations
Theoretical Framework: Green’s Function Approach� Electronic excitations are the poles in G(E)
� Natural framework to account for interactions & finite quasiparticle lifetime
� Correlation effects are collected in the still to bedetermined non-local, energy dependentelectron self energy operator
Hedin’s Equations & the GW Approximation (1965)
Physical Ingredients & Practical Considerations for Real Materials& Illustrative Examples (c1990)
EEQP
ΓΓΓΓQPImG
k(E
)
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Physical Theory for the Self Energy ΣΣΣΣ
Standard perturbation expansion in v(r,r’):� First term is the exchange operator from HF theory –
− Exercise: Derive standard HF expression using G0 from independent particles
� Going to higher order convergent only in the (unphysically) high density limit
Natural question: What about screening v?
Thomas-Fermiscreening model
� Short range effective potential in a metal
v
G0
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Exact, Closed Equations Including ΣΣΣΣDerived following Martin & Schwinger: Hedin, 1965
Screened Coulomb interaction:
GW: Throw away the hardest partVertex function:
Polarizability (screening):
Self energy operator:
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GW: Lowest Order Approximation
Vertex function:
====
++++
vW
G
W
�Screening: v(r-r’) ���� W(r,r’; ωωωω) �Full Green’s function lines: G0 ���� G
Relative to HF, what’s new in a nut-shell:
Exchange + correlation
“Random phase approximation”
Hedin, 1965
Self energy operator:
Polarizability (screening):
Screened Coulomb interaction:
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Screened Coulomb Interaction
Self consistent screening response (Hartree level):
Imεεεε−−−−1111(ωωωω)
ωωωω
plasmon
e-hcontinuum
====
++++
Express via dielectric matrix:
Nota Bene: irreducible P hereoften termed P 0 or χχχχ0
With plane-wave basis:
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Anatomy of GW
Fourier transform:
Insert general, spectral representations:− Suppress real-space indices
G
W
screening of exchange Coulomb hole
Two terms in the GW self energy operator:
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Anatomy of GW: QP Approximation for G
Assume independent electron (QP) model for G:− Energy independent effective potential ���� G
GW self energy operator:
Simplify & use definition of B(E):
Note sum on all states
screening of exchange Coulomb hole
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Anatomy of GW: Static Limit
Full COH term:
Static limit of screened interaction (ωωωωp large):
Energy independent COHSEX Approximation: Hedin, 1965
Imεεεε−−−−1111(ωωωω)
ωωωω
plasmon
e-hcontinuum
Polarization energy,electron at r
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Example: GW for Electron Gas
Screened Coulomb interaction: Lindhard εεεε-1(q,ωωωω)
Quasiparticle equation solutions: Planewaves φφφφk(r) ~ eik·r
Hedin & Lundqvist, 1969
A(k,E), r s=5
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Start of Lecture II
Introduction: Electronic Excitations
Theoretical Framework: Green’s Function Approach
Hedin’s Equations & the GW Approximation (1965)
� The Σ=iGW emerges from the iterative solution of a closedset of equations that formally solve the many-body problem
� Compared to HF, there is dynamical screening of the exchange& the polarization energy gain around the added electron (COH)
� In principle, the G & the W that enter are the fully interactingGreen’s function and screened Coulomb interaction
Physical Ingredients & Practical Considerations for Real Materials& Illustrative Examples (c1990)
G
W
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Application to Real Materials: Key Ingredients
“Best” independent electron (QP) model for G:− LDA, Hybrid, COHSEX, …− Iterate on spectrum
Matrix elements of the self energy for target states:− Note Vref may be Vxc in LDA, Hybrid, …
“Best” dielectric matrix in the RPA:− Complete linear response matrix needed, e.g. from DFT− More details later
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Dielectric Matrix: Qualitative Effects
Atomic scale modulation of screening:
� “Local Field” effects
Dynamic screening:� Full w dependence vs
generalized plasmon pole models( ) ( )rrrr ′−≠′ −− 11 , εε
Hybertsen & Louie, PRB, 1986
ωωωω
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Application to Materials: Key Physical Effects
Local fields in screening increase the band gaps –� Valence band states typically in a
different region of the cell from conduction band states
Static COHSEX approximation over estimates band gaps� Short wavelength contributions to COH
term 2x too big
Dynamic renormalization modest, but quantitatively important� Values of Z ~ 0.8 for semiconductors
QP wavefunctions often very close to KS wavefunctions� Enables 1st order treatment of Σ(r,r’;E)
Bulk Silicon
Hybertsen & Louie, PRL, 1985
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Silicon Bandstructure: 1976 to 1986
Chelikowsky & Cohen, PRB, 1976 Hybertsen & Louie, PRB, 1986
The Gold Standard The New Wave
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Broad Applicability …
Semiconductors & Insulators
Louie & Rubio, Handbook of Materials Modeling, Springer, 2005
Surfaces
Si(111):2x1
Northrup, Hybertsen & Louie, PRL, 1991
C60, Molecules, …
… But Significant Challenges
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Flow for GW Calculation
Ground state calculation for physical structure
Input spectrum and wavefunctions from reference H� Often use DFT (LDA, hybrid, …)
� Must calculate a large number of empty states− Much more expensive in computer time than standard ground state
Calculation of the full dielectric screening response� Must include the full matrix up to a cut-off (control for final quality)
� Includes sums on empty states (control for final quality)� Either full frequency dependence, or input to a plasmon pole model
� Typically scales as N^4 (number of atoms)
Calculation of QP energy corrections from matrix elements of ΣΣΣΣ� Includes sums on empty states (control for final quality)
� Scales with number QP energies needed (more for any type of self consistency)
� Scaling w/ system size varies, but like N^4 to support self consistency− QP wavefunctions needed
self
cons
iste
ncy
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Outline for Lecture II
GW: Physical Ingredients & Practical Considerations for Real Materials& Illustrative Examples (c1990)� “Best G”-”Best W” approach
� Key role for local fields and dynamical corrections
� GW “works” for many materials at this level of implementation� High cost: system size scaling; the necessity to converge sums on empty states
Background: Collective & Optical Excitations
Theoretical Framework: Bethe-Salpeter Equation
BSE: Illustrative Examples for Specific Materials
Cutting-Edge Issues for GW/BSE Theory
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Introduction to Collective Excitations
+++ −−−−
−−−−−−−−
x
σ=nex
ωpElementary argument:
� Electric field:
� Restoring force:
� Oscillation freq:
Physical probe: Energy loss spectra for fast, charged particles
Q
Simple relationship to the dielectric function:
� Macroscopic screening function:
� Density response:
− Unforced oscillations at zeros of εεεεM(q,ωωωω)
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Examples
-40
0
40
0 5 10 15 20 250
20
40
ωωωω (eV)
Re(
εε εε(q,
ωω ωω))
Im( εε εε
−− −− 11 11(q
, ωω ωω))
Lindhard: q=0.2kF at rs=2
broadenedfor display
Bulk Silicon
Philipp & Ehrenreich, Phys Rev, 1963
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Independent Electron Model: Absorption
RPA express, irreducible polarizability (solid):
� Macroscopic εM includes local fields from matrix inversion:
k
Imaginary part corresponds exactly to electron-hole generation rate(optical absorption in the q����0 limit)
� Note: subtlety of longitudinal versus transverse response(the same for cubic crystals)
E
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Independent Electron Model: Absorption
Illustration of Local Fields:� Local polarization response to a
uniform applied E-field
Hanke & Sham, PRL, 1979
Exciton effects are missing –� Shape / oscillator strength --
semiconductor optical spectra
Bulk Silicon
E-f
ield
Hybertsen & Louie, PRB, 1987
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Outline
GW: Physical Ingredients & Practical Considerations for Real Materials& Illustrative Examples (c1990)
Background: Collective & Optical Excitations� Zeros of the macroscopic dielectric function � collective excitations (plasmons)
� Imaginary part of the macroscopic dielectric function � particle-hole excitations− Exciton (electron-hole interaction) effects missing from RPA
Theoretical Framework: Bethe-Salpeter Equation
BSE: Illustrative Examples for Specific Materials
Cutting-Edge Issues for GW/BSE Theory
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BSE Resources
Some key literature references� Elliott, Phys Rev 108, 1384 (1957).
− Classic treatment of excitons at the band edge of semiconductors
� Sham & Rice, Phys Rev 144, 708 (1966)− The first bridge between BSE and the effective-mass treatment of excitons
� Del Sole & Fiorino, Phys Rev B 29, 4631 (1984)− Sorts out the longitudinal versus transverse field issue & clarifies that the local fields are properly included in the widely used BSE expression
� Strinati, Phys Rev B 29, 5718 (1984)− Concise exposition of the basic many-body expressions leading up to the BSE
� Rohlfing & Louie, Phys Rev B 62, 4927 (2000)− Clear exposition of the implementation of BSE
� Onida, Reining and Rubio, Rev. Mod. Phys, vol. 74, pp. 601-659, 2002− Includes both GW and BSE; includes TD-DFT; materials examples and exposition emphasize optics
Older book:� R.S. Knox, Theory of Excitons, Solid State Physics Supplement Vol 5, 1963
− More physical exposition, including TD-HF
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Vertex Corrections: Electron-Hole Interactions
Recall the vertex function from Hedin’s closed equation set:
Approximate from GW:
Simplified self-consistent vertex equation:
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Vertex Corrections: Electron-Hole Interactions
Simplified self-consistent vertex equation:
Iterate to see the structure:
3
1
2
3
1
2
3
6
7
1
2
= + + + …ΓΓΓΓ
Note: stop doubling all the G & W lines!
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Vertex Corrections to Polarization: Ladder Diagrams
Incorporate into the polarization:
= + + + …
Which goes into the final screened Coulomb interaction (dielectric function):
= +
+ + …
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Solution Strategy: Spectral Representation in e/h Pairs
Generalize to represent part of the two particle Green’s function that satisfies the BSE integral equation:
Graphical schematic for the BSE:
3
4
6
5“exchange”
6
5
3
4screened e/h
1 21
1’
2’
2
1
1’
2’
2
1
1’
2’
2
6
5
2’
2
1
1’
3
4
L LL0 L0= +
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General BSE Expressions
Electron/hole basis set for homogeneous equation:
Comments:� Resonant & anti-resonant terms coupled� Frequency self consistency required if
dynamical screened interaction retained
Full BSE equations:
Notation following Rohlfing & Louie, PRB, 2000
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Widely Used Simplifications
Assume KAB small & decouble A/B to have a single eigenvalue equation� Tamm-Dancoff approximatio (commonly used in TD-DFT also)
Assume static screening only
Restrict to zero center of mass momentum excitons
Final optical response function (absorption):
E
k
e/h exchange
screened e/hattraction
Nota Bene: matrix element includes
coherent exciton effects
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Outline
GW: Physical Ingredients & Practical Considerations for Real Materials& Illustrative Examples (c1990)
Background: Collective & Optical Excitations
Theoretical Framework: Bethe-Salpeter Equation� Start from GW input quasiparticle energies
� BSE derived equations of motion for excitons that include screened e/h attraction and bare e/h exchange
� Direct connection to optical absorption including local field effects
BSE: Illustrative Examples for Specific Materials
Cutting-Edge Issues for GW/BSE Theory
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Example: Bulk GaAs
Rohlfing & Louie, PRL, 1998; PRB, 2000
Expt
No e/hBSE
Basis set:� (3 val)X(6 cond)X(500 k) = 9000 fcns
− Energy spacing about 0.15 eV
� Matrix element (KAA,d, KAA,x) dominate− Interpolation scheme used
Dramatic change in oscillator strength:� NOTE: In the continuum (above gap),
states do NOT shift:− Spectral weight (matrix elements) change due to electron-hole correl.
Bound exciton states appear in the gap with scale ~ meV:
� Requires ~1000 k-points near Γ to resolve the Wannier excitons in k-space
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Exciton Binding & Character in Organic Crystals
Tiago, Northrup & Louie, PRB, 2003Hummer, Puschnig & Ambrosch-Draxl, PRL, 2004
singlet singlet
triplet
Anthracene Pentacene
Singlet: 0.64 eVTriplet: 1.86 eV
Singlet: 0.3 eVTriplet: 1.1 eV
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Rutile TiO2 Optical Spectra
GW/BSEExpt
Expt: Cardona and Harbeke, Phys Rev 137, A1467, 1965
Kang & Hybertsen, Phys Rev B 82, 085203, 2010
Neglect of e-phonon interaction:� Lowest (dipole dark) exciton 0.2 eV too
high compared to spectroscopy
� Exciton binding scale much too big
Oscillator strength issue near 8 eV:� Other oxides: Schleife, et al, PRB, 2009
� Tamm-Dancoff issue?� Experimental analysis?
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Key Materials Challenges for MBPT
Application to complex bulk solids, point defects, & heterogeneous interfaces
� Will MBPT be a useful tool for materials discovery ?
Need & utility for a calibrated, static model that goes beyond hybrid functionals,but with no explicit sums on empty states
Fundamental investigation of the impact of electron-phonon coupling on quasiparticle & optical excitations in titinates & related
� Classic example of intermediate to strong coupling
GaN
H2O
Shen, Small, Wang, Allen, Fernandez-Serra, Hybertsen, & Muckerman, J Phys
Chem C 114, 13695, 2010
Pascual, Camassel and Mathieu,
Phys Rev Lett, 1977; Phys Rev B, 1978
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Outline
GW: Physical Ingredients & Practical Considerations for Real Materials& Illustrative Examples (c1990)
Background: Collective & Optical Excitations
Theoretical Framework: Bethe-Salpeter Equation
BSE: Illustrative Examples for Specific Materials� Tamm-Dancoff + static screening remarkably successful for optical absorption
� Challenges with BZ sampling and other convergence
Cutting-Edge Issues for GW/BSE Theory
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Example: Monoclinic VO2 Band Gap
T > 340 K: Metallic Rutile
T < 340 K: Insulating Monoclinic
Eyert, Ann Phys, 2002Gatti, Bruneval, Olevano & Reining, PRL, 2007
LDA: Ground state structure
GW: QP energies –Self consistent φk (COHSEX level)
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GW: To Be Self Consistent … or Whether Tis Nobler …
Hedin’s deriviation: Dressed G x Dressed W
In Baym-Kadinoff theory: GW is a conserving approximation when full self consistent� Charge is conserved, etc.
Electron gas studies− Holm & von Barth, PRB, 1998; 1999
� The notation G0W0, GW0, etc, refers to which component is at least parially self consistent
� Self consistent, GW gives excellent total energies
� Self consistent, GW gives unphysical spectral functions− Note: Unlike the total energy, there are no ‘numerically exact’ results for A(E)
Applications to real materials – “Quasiparticle selfconsistency”− Kotani, van Schilfgaarde & Faleev, PRB, 2007
� Qualitative arguments: Self consistency without vertex corrections unphysical
� Concrete proposal for a ‘best’ Veff derived from QP part of Σ− Most widely used type of self consistency, generally increasing gaps
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Impact of Self Consistency
Shishkin, Marsman & Kresse, PRL, 2007
Van Schilfgaarde, Kotani& Faleev, PRL, 2006
fxc in W only
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ZnO: The Bete Noir of GW
Shih, et al, PRL, 2010
Stankovski, et al, PRB, 2011
Numerical Convergence Model for Dynamic Screening
Common example arguing forself consistency, …
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Reflections
Why did GW emerge in the 1980’s ?� Reliability of electronic structure methods (pseudopotential & other)
� The relative simplicity of GW in a planewave basis & the ability to numerically converge the calculations for basic materials
� The rapid validation by a second, independent group (Godby, Schluter & Sham)
� Convincing evidence that the ‘band-gap’ problem in DFT was real� For many materials, “Best G, Best W” approach is adequate
Why do we ask “Which GW” in the 2010’s ?� Struggles with numerical convergence, particularly with respect to empty states
� On-going dialogue between pseudopotential & all-electron methods, particularly around the important role of “n-1” shell core levels
� The real need for a physical control of the input electronic structure: Materials where KS wavefunctions are not a good approximation to QP wavefunctions
� More generally, the drive for a theory that is independent of DFT input …… or more generally does not depend on the initial guess …
Today “GW/BSE” is a vibrant field with many important groups contributing to solve big challenges