Quasiparticle and Optical Calculations of Low-Dimensional Systems
Sahar SharifzadehMolecular Foundry, LBNL
Phenomena Necessitating Explicit Calculations of Low-Dimensional Systems
Mosconi, et al J. Am. Chem. Soc. (2012)Stanford.edu
Unique Electronic and Optical Properties for Reduced Dimensionality
1D 2D0DSharifzadeh, et al Euro Phys J. B 85 323 (2012)
• Ionization energies• Energy level alignment and charge transfer at surfaces• Confinement effects on charged and optical excitations
Neaton, Hybertsen, Louie,PRL. 97, 216405 (2006)Spataru, et al PRL 077402 (2004)
Computational Challenges with Reduced Dimensionality
• Aperiodicity within periodic boundary conditions• Truncation of Coulomb potential• Convergence behavior of self-energy
1D 2D0DSharifzadeh, et al Euro Phys J. B 85 323 (2012)
Neaton, Hybertsen, Louie,PRL. 97, 216405 (2006)Spataru, et al PRL 077402 (2004)
Aperiodicity within periodic boundary conditions
• Large amount of vacuum between periodic images along aperiodic direction– Keep periodic images from interacting
• Determining the lattice vector length– Increasing until results no longer change is too expensive within GW/BSE– Decide unit cell size based on charge density distribution
Charge density is contained within ½ of lattice vector length
99% of charge-density
Aperiodicity and periodic boundary conditions
•Vacuum level correction at the DFT level– Kohn-Shame eigenvalues shifted by potential at the edges of cell– Correct by the average electrostatic energy along faces of supercell
(Hartree and electron-ion)
Truncation of Coulomb potential• GW and BSE utilize the Coulomb and screened Coulomb interaction
• Long-range interactions make it computationally infeasible to increase lattice vectors until periodic images do not interact
Truncation Schemes within BerkeleyGW• Cell box: 0D• Cell wire: 1D• Cell slab: 2D• Spherical: Define radius of truncation
• Cell truncation: at half lattice vector length– Analytical form for Coulomb potential in k-space
• Spherical truncation: convenient, available in many packages
cVW 1
Example, Cell Slab Truncation
Z
||)2/(
rr Lz )(VC ))2/(cos(1(
||4 )2/(
2 Lke zLkxy k
)(VC k
Example, GaN sheets
QuasiparticleCorrection Separation length (a.u.)
12x12x12 k-mesh 10 14 18
No Truncation 1.98 2.05 2.10
Truncation 2.53 2.55 2.58
Ismail-Beigi PRB 73 233103 (2006)
• Convergence improved with truncation
Example, Cell Slab Truncation: q 0 Singularity
When computing the self-energy• Average Vc for q-points near zero high sensitivity to k-point mesh• Average
– Increased stability – Better convergence
Pick, Cohen, and Martin PRB 1 910 (1970)
)0()0( 100 qq cVW
0)( q as diverges Vc q
2
100
||)(
)()(1)0(
qqq
f
fVc
Ismail-Beigi PRB 73 233103 (2006)
For each system, * We compute for small q average W
Example: GaN 2D Sheets
• No truncation: long-range interactions make convergence difficult• No truncation: increase in size of vacuum requires increase of k-
point mesh (need uniform k-point mesh)• With truncation, W averaging improves convergence
K-point Mesh
No Truncation
Truncation Average V
Truncation Average W
4x4x1 1.55 3.53 2.37
8x8x1 1.73 2.09 2.48
14x14x1 2.03 2.56 2.49
18x18x1 2.52 2.49
GW correction to LDA gap at G (eV)
Convergence Behavior of Self-Energy for Absolute Quasiparticle Energies
• Convergence parameters– Number of bands (Nc) – can be
different for and S– Dielectric G-vector cutoff (cutoff)
• Challenging for absolute energies– Parameters are inter-dependent– Converge very slowly
CHXSXX SSSS
Dielectric matrix and unoccupied states
Depends ondielectric matrix
Sharifzadeh, Tamblyn, Doak, Darancet, Neaton Euro Phys J. B 85 323 (2012)
Exact static result
Slow Convergence of Self-Energy
• Slow convergence of Coulomb-hole term• Static remainder approach to complete S
Deslippe, Samsonidze, Jain, Cohen, Louie, PRB 87 165124 (2013)
Example: Ionization Energies and Electron Affinities of Small Molecules
• cutoff and Nc are interdependent• Absolute eigenvalues converge much more slowly than energy differences• This will be very challenging when studying level alignment at interfaces
Converged Eigenvalues Agree Well with Experiment but Can Differ with Other GW Packages
Calculated IP (eV) BEN TP BDA
BerkeleyGWa
(G0W0@PBE) 9.4 9.0 7.3
IP experiment 9.2 8.9 7.3FHI-AIMSb
(G0W0@LDA,PBE) 8.9,9.0 ---- ----
FIESTAc
(G0W0@LDA) 9.0 8.4 ----
GWLd
(G0W0@LDA) 9.1 ---- ----
(G0W0@PBE)e 9.2 6.9
Different approximations can lead to slightly different results• Basis setso Planewaveso Atom-centered
• Frequency-dependenceo Plasmon-pole models o Full frequency approaches
• Description of the core
a) Sharifzadeh, et al, EPJB 85, 323 (2012)b) Ren, et al New J. Phys. 14, 053020 (2012);
Marom, et al. PRB (2012)c) Blase, Attaccalite, Olevano PRB 83, 115103 (2011)d) Umari, Stenuit ,Baroni PRB 79, 201104R 2009e) Pham, et al PRB 87, 15518 (2013)
Example: How Do Nature and Energy of Excitations of an Organic Molecule Change with Phase?
• Comparison of calculation with surface-sensitive photoemission expts.• Design of molecules with certain properties valid in the solid-state
Gas-phase Bulk crystal 1-layer slab
Convergence parameters• E(Nc) = 2.6 Ry (35 eV) – good for energy differences
Number of bands: 3200 for molecule, 600 for bulk crystal, 900 for surface • K-points in the solid: 4x4x2 (bulk); 4x4x1 (slab)
Solid-State Polarization Dominates Change in Energetics
4.5 eV 2.2 eV
IP
EA
+P
-P
2.6
Gas-phase Bulk crystal 1-layer slab
R
ε
Sharifzadeh, Biller, Kronik, Neaton, PRB 85, 125307(2012)
R
ε
R
qP2
)1(2 Gap= 2.1 eV Gap = 2.3 eV
Pentacene: Singlet and Triplet Excitations
2 Å 8 Å
rrdrFr 3)( Average electron-hole distance
Triplet Singlet
1.2 1.75 0.7 2.2
Crystal:Molecule:
ExcitationEnergy (eV)
* 10x10x6 k-mesh
Svc
SScv
cv
ehSvc
QPv
QPc AAcvKvcAEE
'')(
''''1''12
ccWvvcvrr
vccvKvc eh
Direct termExchange term
Triplet
Sharifzadeh, Darancet, Kronik, Neaton, J. Phys. Chem Lett 2013; Cudzzo et al PRB (2012);Tiago, Northrup Louie, PRB (2003).
d|),(|)(F h32
hh rrrrr
Example: BandStructure and Optical Excitations in Metallic Carbon Nanotubes
~40 x ~40 x 5 a.u.3
60 Rydberg Wavefunction Cutoff6 Rydberg Dielectric Cutoff
~1000 Bands1x1x32 coarse, 1x1x256 fine
Slide from Jack Deslippe, NERSC
Bandstructure of Metallic Carbon Nanotubes
1.47eV
Optically Forbidden
Optically Allowed
Due to symmetry have optical gap.
Metallic screening usually prohibits bound excitonic states.
Slide from Jack Deslippe, NERSC
(10,10) SWCNT Band Structure
Excitons in Metallic Tubes
.06 eV
•Peak from a single eigenvalue.
•Exciton binding energy - 0.06 eV.
•The onset is calculated to be 1.84 eV.
Experimental value*: 1.89 eV
(10,10)
(12,0)
Slide from Jack Deslippe, NERSC
(Experiment) Fantini, Jorio, Souza, Strano, Dresselhaus, Pimenta, Phys. Rev. Lett. 93, 147406 (2004)
(Theory) Deslippe, Prendergast, Spataru, Louie, Nano Lett. 7 1626 (2007)
Summary
• Computational challenges with low-dimensional materials• BerkeleyGW offers methodological developments to help
overcome these challenges– Truncation of Coulomb potential– Averaging of screened Coulomb potential at q 0– Static remainder approach to complete self-energy
• Convergence is important for absolute energies