CECAM, Berlin, 8/4&6/12 1 Theoretical Framework for Electronic & Optical Excitations, the GW & BSE Approximations and Considerations for Practical Calculations Mark S Hybertsen Center 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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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.
CECAM, Berlin, 8/4&6/12 2
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 …
� Efficiency: Complexity one order higher than ground state (at least)
CECAM, Berlin, 8/4&6/12 3
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
CECAM, Berlin, 8/4&6/12 4
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)
CECAM, Berlin, 8/4&6/12 6
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
CECAM, Berlin, 8/4&6/12 7
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
CECAM, Berlin, 8/4&6/12 8
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
CECAM, Berlin, 8/4&6/12 9
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
Louie & Rubio, Handbook of Materials Modeling, Springer, 2005
Surfaces
Si(111):2x1
Northrup, Hybertsen & Louie, PRL, 1991
C60, Molecules, …
… But Significant Challenges
CECAM, Berlin, 8/4&6/12 35
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
CECAM, Berlin, 8/4&6/12 36
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
CECAM, Berlin, 8/4&6/12 37
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)
CECAM, Berlin, 8/4&6/12 38
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
CECAM, Berlin, 8/4&6/12 39
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
CECAM, Berlin, 8/4&6/12 41
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
CECAM, Berlin, 8/4&6/12 42
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
CECAM, Berlin, 8/4&6/12 43
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!
CECAM, Berlin, 8/4&6/12 45
Vertex Corrections to Polarization: Ladder Diagrams
Incorporate into the polarization:
= + + + …
Which goes into the final screened Coulomb interaction (dielectric function):
= +
+ + …
CECAM, Berlin, 8/4&6/12 46
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= +
CECAM, Berlin, 8/4&6/12 47
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
CECAM, Berlin, 8/4&6/12 48
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
CECAM, Berlin, 8/4&6/12 49
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
CECAM, Berlin, 8/4&6/12 50
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
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
CECAM, Berlin, 8/4&6/12 57
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, …
CECAM, Berlin, 8/4&6/12 59
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