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.


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)


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


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)


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)


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


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



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



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


Hartree-Fock (Mean-Field) Theory

Condition on the spin-orbitals to optimize the ground-state energy:

Postulate a variational wavefunction: Slater determinant form –



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



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


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


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



Physical Ingredients & Practical Considerations for Real Materials& Illustrative Examples (c1990)


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


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


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:


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


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)|


Outline for Today


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



EEQP

ΓΓΓΓQPImG

k(E

)


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


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:


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:


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:


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:


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


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


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


Start of Lecture II




� 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


G

W


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


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


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


Silicon Bandstructure: 1976 to 1986

Chelikowsky & Cohen, PRB, 1976 Hybertsen & Louie, PRB, 1986

The Gold Standard The New Wave


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


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


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


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)


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


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


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


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





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


Vertex Corrections: Electron-Hole Interactions

Recall the vertex function from Hedin’s closed equation set:

Approximate from GW:

Simplified self-consistent vertex equation:


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!


Vertex Corrections to Polarization: Ladder Diagrams

Incorporate into the polarization:

= + + + …

Which goes into the final screened Coulomb interaction (dielectric function):

= +

+ + …


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= +


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


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


Outline



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




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


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


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?


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


Outline




BSE: Illustrative Examples for Specific Materials� Tamm-Dancoff + static screening remarkably successful for optical absorption

� Challenges with BZ sampling and other convergence



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)


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


Impact of Self Consistency

Shishkin, Marsman & Kresse, PRL, 2007

Van Schilfgaarde, Kotani& Faleev, PRL, 2006

fxc in W only


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, …


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

Theoretical Framework for Electronic & Optical Excitations,...

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