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Quasiparticle and Optical Calculations of Low-Dimensional Systems

Quasiparticle and Optical Calculations of Low-Dimensional Systems. Sahar Sharifzadeh Molecular Foundry, LBNL. Phenomena Necessitating Explicit Calculations of Low-Dimensional Systems. Mosconi, et al J. Am. Chem. Soc. (2012). Stanford.edu.

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Quasiparticle and Optical Calculations of Low-Dimensional Systems

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  1. Quasiparticle and Optical Calculations of Low-Dimensional Systems Sahar Sharifzadeh Molecular Foundry, LBNL

  2. Phenomena Necessitating Explicit Calculations of Low-Dimensional Systems Mosconi, et al J. Am. Chem. Soc. (2012) Stanford.edu

  3. Unique Electronic and Optical Properties for Reduced Dimensionality • Ionization energies • Energy level alignment and charge transfer at surfaces • Confinement effects on charged and optical excitations 0D 1D 2D Sharifzadeh, et al Euro Phys J. B 85 323 (2012) Neaton, Hybertsen, Louie, PRL. 97, 216405 (2006) Spataru, et al PRL 077402 (2004)

  4. Computational Challenges with Reduced Dimensionality • Aperiodicity within periodic boundary conditions • Truncation of Coulomb potential • Convergence behavior of self-energy 0D 1D 2D Sharifzadeh, et al Euro Phys J. B 85 323 (2012) Neaton, Hybertsen, Louie, PRL. 97, 216405 (2006) Spataru, et al PRL 077402 (2004)

  5. 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

  6. 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)

  7. 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

  8. Example, Cell Slab Truncation Example, GaN sheets Z • Convergence improved with truncation Ismail-BeigiPRB73 233103 (2006)

  9. 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 For each system, * We compute g for small q average W Ismail-BeigiPRB73 233103 (2006) Pick, Cohen, and Martin PRB1 910 (1970)

  10. Example: GaN 2D Sheets GW correction to LDA gap at G (eV) • 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

  11. Convergence Behavior of Self-Energy for Absolute Quasiparticle Energies • Convergence parameters • Number of bands (Nc) – can be different for e and S • Dielectric G-vector cutoff (ecutoff) • Challenging for absolute energies • Parameters are inter-dependent • Converge very slowly Depends on dielectric matrix Dielectric matrix and unoccupied states Sharifzadeh, Tamblyn, Doak, Darancet, Neaton Euro Phys J. B 85 323 (2012)

  12. Slow Convergence of Self-Energy • Slow convergence of Coulomb-hole term • Static remainder approach to complete S Exact static result Deslippe, Samsonidze, Jain, Cohen, Louie, PRB87 165124 (2013)

  13. Example: Ionization Energies and Electron Affinities of Small Molecules • ecutoff and Nc are interdependent • Absolute eigenvalues converge much more slowly than energy differences • This will be very challenging when studying level alignment at interfaces

  14. Converged Eigenvalues Agree Well with Experiment but Can Differ with Other GW Packages • Different approximations can lead to slightly different results • Basis sets • Planewaves • Atom-centered • Frequency-dependence • Plasmon-pole models • Full frequency approaches • Description of the core Sharifzadeh, et al, EPJB85, 323 (2012) Ren, et al New J. Phys. 14, 053020 (2012); Marom, et al. PRB (2012) Blase, Attaccalite, Olevano PRB83, 115103 (2011) Umari, Stenuit ,Baroni PRB79, 201104R 2009 Pham, et al PRB87, 15518 (2013)

  15. 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)

  16. R R ε ε Solid-State Polarization Dominates Change in Energetics EA -P 2.2 eV 4.5 eV 2.6 IP +P Gap = 2.3 eV Gap= 2.1 eV Gas-phase Bulk crystal 1-layer slab Sharifzadeh, Biller, Kronik, Neaton, PRB 85, 125307(2012)

  17. Pentacene: Singlet and Triplet Excitations Average electron-hole distance Singlet Triplet Triplet 2 Å 8 Å Excitation Energy (eV) Crystal: Molecule: 1.2 1.75 0.7 2.2 Exchange term Direct term Sharifzadeh, Darancet, Kronik, Neaton, J. Phys. ChemLett2013; Cudzzo et al PRB (2012); Tiago, Northrup Louie, PRB (2003). * 10x10x6 k-mesh

  18. Example: BandStructure and Optical Excitations in Metallic Carbon Nanotubes ~40 x ~40 x 5 a.u.3 60 Rydberg Wavefunction Cutoff 6 Rydberg Dielectric Cutoff ~1000 Bands • 1x1x32 coarse, 1x1x256 fine Slide from Jack Deslippe, NERSC

  19. Bandstructure of Metallic Carbon Nanotubes (10,10) SWCNT Band Structure Optically Forbidden Optically Allowed 1.47eV Due to symmetry have optical gap. Metallic screening usually prohibits bound excitonic states. Slide from Jack Deslippe, NERSC

  20. Excitons in Metallic Tubes • 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) .06 eV (12,0) (Experiment) Fantini, Jorio, Souza, Strano, Dresselhaus, Pimenta, Phys. Rev. Lett. 93, 147406 (2004) (Theory) Deslippe, Prendergast, Spataru, Louie, Nano Lett. 7 1626 (2007) Slide from Jack Deslippe, NERSC

  21. 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

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