GW approximation and its implementation in VASP

GW-ApproximationRAJASEKARAKUMAR VADAPOO

Department of Physics, University of Puerto Rico-RP, PR-00925, USA.

Thanks to:

Sridevi Krishnan, Hulusi Yilmaz, Carlos Marin, Julian Velev

More info: http://nanophysics.wordpress.com/

Why we need of GW?

� DFT based on Hohenberg- Kohn, Kohn-Sham theorem.

� DFT- natural choice to find the groundstate properties of the many electron system.

� Excited states properties – functionals of the ground state density as well.

� Hohenberg- Kohn theorem- doen’t provide explicit mathematical form for excited

states

� Excited states properties can be accessed more directly from Many body

perturbation theory.

� In practice its implemented within GWA for the electronic self energy based on

- perturbative evaluation of Kohn-Sham orbitals.

What is GW Approximation (GWA)?

• GWA- relies on perturbative treatment starting from DFT.

• GWA- systematic algebraic approach on the basis of Green fn. Techniques.

• Many body perturbation theory- a method to determine quasiparticle excitations in solids.

• Main ingredient: self-energy.

- contain many-body exchange and correlation effects beyond Hartree

potential.

• GWA- constitutes an expansion of self-energy upto linear order in screened coulomb potential

– Describes the interaction between quasiparticles & includes dynamic screening through the creation of exchange-correlation holes around the bare particles

Bandgap by DFT & GW

Christoph Friedrich and Arno Schindlmayr, Computational Nanoscience:Do it yourself!-Vol.31

How we measure bandgap

experimentally?

�Strong Coulomb interaction questions single electron concepts: band structure or Fermi surface

� At least we can retain nearly-independent-particle picture

- if we consider quasiparticles instead of electron/ hole.


What is Quasiparticle?


� Coulomb hole reduces the total charge of the quasiparticle

� The effective interaction between quasiparticles is screened & considerably

weaker than the bare coulomb interaction between electrons.

� screened interaction is small so that quasiparticle is almost independent

� Justify independent particle approximation- Success of mean field theories.

N & (N+/-1) particles- explain by many body perturbation theory.

G(rt, r’t’) : contains excitatin energy (εi ), excitation lifetime

Now E[G] which contain more than electron densitycontrast to the E[n]

How GW-implemented in VASP?

Quasi particle Energy in the GW calculation calculated by:

Where,

� T – Kinetic energy operator

� Vn-e – The potential of the nuclei

� VH – Hatree potential

� Σ(Enk ) – Self energy term

� n – band index

� k – k-point index (reciprocal space)

Self consistent GW

Self energy matrix

Where,

� W – Dynamically screened potential calculated in the common random

phase approximation (RPA)

� µ - Fermi Energy

When, GGA wavefns are close to GW one :

-Non diagonal element of self energy matrix

could set as “0”

Updating the quasiparticle energy

Where,

Znk – renormalization factor

M. Shishkin and G. Kresse, PRB, 75, 235102 2007

GW0

Perform single shot

Single electron

energy

Input

G0W0

GW

Iterate with fixed W Update G & WRPA

How generalized Kohn-Sham eigen

values corrected using green fn.

G0W0 –QP shift for a certain gKS state is

GW self energy

Nonlocal exachange

Correlation

potential

Screened Coulomb kernelShort range wt.Inverse screening length

F. Fuchs, J. Furthmüller, and F. Bechstedt,M. Shishkin and

G. Kresse, PRB, 76, 115109 2007

F. Fuchs, J. Furthmüller, and F. Bechstedt,M. Shishkin and

G. Kresse, PRB, 76, 115109 2007

DFT & GW- comparison


�GGA- consistent GGA treatment

of valence & core electrons

� GW- core-valence interaction is re-

Evaluated on the Hatree-Fock level.

GW-error Vs RPA-error


Cont…

� Errors are larger for materials

With semicore d-states

� Is it due to LDA doesn’t cancel

The Couloumb self interaction within

d-shell completely?

� DFT+U approach worked well for

GW0- for GaAs, GaN but

Too large bandgap for ZnO.

� So, worked for the material d-shell

located Well below the valence band.Transition metal oxide & Rare-earth oxide

Not well described in LDA/GGA wavefn.

So, not described in GW calc. by KRESSE group.


Is GW correction gives exact bandgap?

Not for all materials!!! then what is the exact problem?

� G0W0 : large errors for systems with shallow d states such as GaAs, ZnO, CdS.

�GW0: The above problem partially remedied by iterating one electron energies in the Green

fn : underestimate bandgap 15% for the above listed materials.

�GW : Overestimate the bandgap

� scGW ( self consistent quasiparticle GW) – avoided the loss of intensity due to

quasiparticle peaks to satellites.

�This might be due to the neglect of the attractive interaction between electrons & holes,

which is responsible for the excitonic features in the absorption spectra.

How to solve it?

� Recently proposed Vertex correction to the scGW calculation

Vertex correction.

M. Shishkin, M. Marsman, and G. Kresse, PRL, 99, 246403 (2007)

Inorder to determine quasiparticle peaks once need to determine all solution of a nonlinear

one-electron–like Schro¨dinger equation:

Where,

T - the kinetic energy operator

V- corresponds to the electrostatic potential of the nuclei and electrons

-self-energy is energy dependent and nonlocal

Linearization around some reference energy for state n and iteration i as follows:

Recasting the above equation becomes

-genealized and non-Hermitian (but linear ) eigen value problem

Quasiparticle peaks are not the one normalized which reflected by the presence of ovelap

Operator (S) – 75% intensity normalized & 25% in satellites ( which lack after linearlization)

Possible route: Determine Hamiltonian (H) & Overlap operator (S) in a suitable basis set

[(ex: DFT wavefn.) ]

Where,

= Unitary matrix

= diagonal eigen value matrix

ISSUE:1

ISSUE:2Inclusion of electron-hole interaction in the dielectric matrix

The screened Coulomb kernel In time dependent DFT, the full polarizability (χ) given by Dyson-like equation:

Where,

Independent particle polarizability

Coulomb

kernel

ν=

Local density functional:


How to do G0W0-calculation in VASP

SYSTEM=Si

NBANDS=150

ISMEAR=0

SIGMA=0.05

LOPTICS= .TRUE.

Step:1

WAVEDER (derivative of wave fn.)

SYSTEM=Si

NBANDS=150

ISMEAR=0

SIGMA=0.05

LOPTICS= .TRUE.

ALGO= GW0

NOMEGA= 50

~

Step:2 for G0W0

INCAR

INCAR

copy WAVEDER from step:1

Don’t copy WAVECAR from step:1 CHGCAR

Go for

DOS & BAND

How to do GW-calculation


Don’t copy WAVECAR from step:1

CHGCAR

Go for

DOS & BAND

Step:2

Or INCAR

Call VASP

repeatedly

�

How to do GW0-calculation


Don’t copy WAVECAR from step:1

CHGCAR

Go for

DOS & BAND

Or

�

LSPECTRAL=.T.

Results:

GGA : 0.67 eV (0.62 ev)

G0W0 : 0.9 eV (1.12 eV)

GW : 1 eV (1.28 eV)

GW0 : 2.86 eV (1.2 eV)

GW approximation and its implementation in VASP

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