GW approximation and its implementation in VASP
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Transcript of 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.
Christoph Friedrich and Arno Schindlmayr, Computational Nanoscience:Do it yourself!-Vol.31
What is Quasiparticle?
Christoph Friedrich and Arno Schindlmayr, Computational Nanoscience:Do it yourself!-Vol.31
� 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
M. Shishkin and G. Kresse, PRB, 75, 235102 2007
�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
M. Shishkin and G. Kresse, PRB, 75, 235102 2007
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.
M. Shishkin and G. Kresse, PRB, 75, 235102 2007
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:
M. Shishkin, M. Marsman, and G. Kresse, PRL, 99, 246403 (2007)
M. Shishkin, M. Marsman, and G. Kresse, PRL, 99, 246403 (2007)
M. Shishkin, M. Marsman, and G. Kresse, PRL, 99, 246403 (2007)
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
copy WAVEDER from step:1
Don’t copy WAVECAR from step:1
CHGCAR
Go for
DOS & BAND
Step:2
Or INCAR
Call VASP
repeatedly
�
How to do GW0-calculation
copy WAVEDER from step:1
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)