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(

qq

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