GW and Bethe-Salpeter Equation Approach to Spectroscopic Properties. Steven G. Louie

Transcription

1 GW and Bethe-Salpeter Equation Approach to Spectroscopic Properties Steven G. Louie Department of Physics, University of California at Berkeley and Materials Sciences Division, Lawrence Berkeley National Laboratory Supported by: National Science Foundation U.S. Department of Energy

2 First-Principles Study of Material Properties = igw Fermi sea + (excitonic) Fermi sea

3 Content Quasiparticle excitations - The GW approximation - Applications to solids, surfaces and nanostructures Excitons, optical response, and forces in the excited state - The Bethe-Salpeter Equation - Applications to crystals, surfaces, nanotubes, selftrapped excitons Some more-correlated systems

4 Quasiparticle Excitations Kohn-Sham Eigenvalues QP Energies One simple example: the Homogeneous Interacting Electron System Standard K-S equation: V ext +V H + E xc (r)= KS (r) (r) V ext + V H = constant V xc (r) = E xc (r) constant Free electron dispersion (m* = m e, infinite lifetime, etc.) WRONG!

5 Additional Theoretical Issues Kohn-Sham formulation is only one approach to DFT. - not unique - different formulation different eigenvalues How shall we interpret the K-S eigenvalues? - electron addition energies? - optical transition energies?

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7 Diagrammatic Expansion of the Self Energy in Screened Coulomb Interaction

8 H = H o + (H - H o ) Hybertsen and Louie (1985)

9 Quasiparticle Band Gaps: GW results vs experimental values Materials included: InSb, InAs Ge GaSb Si InP GaAs CdS AlSb, AlAs CdSe, CdTe BP SiC C 60 GaP AlP ZnTe, ZnSe c-gan, w-gan InS w-bn, c-bn diamond w-aln LiCl Fluorite LiF Compiled by E. Shirley and S. G. Louie

10 Quasiparticle Band Structure of Germanium Theory: Hybertsen & Louie (1986) Photoemission: Wachs, et al (1985) Inverse Photoemission: Himpsel, et al (1992)

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12 Self-energy Corrections in Graphene Nanoribbons -states -states NFE- sheet states

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14 Si(111) 2x1 Surface Measured values: Bulk-state qp gap Surface-state qp gap Surface-state opt. gap 1.2 ev 0.7 ev 0.4 ev

15 Rohlfing & Louie PRL,1998.

16 Optical Absorption Spectrum of SiO2 Chang, Rohlfing& Louie. PRL, 2000.

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18 M. Rohfling and S. G. Louie, PRL (1998)

19 Both terms important! repulsive attractive

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22 Optical Absorption Specturm of GaAs Bound excitons Rohlfing & Louie PRL, 1998.

23 Optical Absorption Spectrum of SiO2 Chang, Rohlfing& Louie. PRL, 2000.

24 Nanostructures Size and restricted geometry => quantum confinement, enhanced many-electron interaction, reduced dimensionality, and symmetry effects Novel properties and phenomena Useful in applications Small can be different! Size Bawendi Group: Colloidal CdSe quantum dots dispersed in hexane.

25 Optical Excitations in Carbon Nanotubes Recent advances allowed the measurement of optical response of well characterized, individual SWCNTs. Response is quite unusual and cannot be explained by conventional pictures. Many-electron interaction (self-energy and excitonic) effects are very important => interesting physics (n,m) carbon nanotube

26 First-principles Study of Optical Properties Many-electron interaction effects - Quasiparticles and the GW approximation - Excitonic effects and the Bethe-Salpeter equation + Single-walled carbon nanotubes - Absorption spectra - Exciton binding energies and wavefunctions - Radiative lifetime,

27 Quasiparticle Self-Energy Corrections (10,10) metallic SWCNT (8,0) semiconducting SWCNT Metallic tubes -- stretch of bands by ~15-25% (velocity renormalization) Semiconductor tubes -- large opening (~ 1eV) of the gap Spataru, Ismail-Beigi, Benedict & Louie, PRL (2004)

28 GW Quasiparticle Band Dispersion of Metallic CNTs Quasiparticle energy corrections: larger compared to graphite increase with increasing diameter E qp (ev) (10,10) Quasiparticle Fermi Velocities (10 6 m/s) LDA QP GW shift (3,3) % (5,5) % (10,10) % Graphene % E LDA -E F

29 Absorption Spectrum of Semiconducting (8,0) Carbon Nanotube Spataru, Ismail-Beigi, Benedict & Louie, PRL 92, (2004) (r e,r h ) 2 (Not Frenkel-like) Long-range attractive electron-hole interaction Spectrum dominated by bona fide and resonant excitons Large binding energies ~ 1eV! [Experimental verification: Wang, Heinz et al, (2005); Ma, Fleming, et al. (2005); Maultzsch, Molinari, et al, (2005), Avouris, et al ]

30 Optical transition energies (in ev) of four semiconducting CNTs 1st transition (E 11 ) 2nd transition (E 22 ) E 22 /E 11 Theory Exp. Theory Exp. Theory Exp. (7,0) a a a (8,0) a a a (10,0) a a a (11,0) b b b a S. Bachilo, et al. (2002), b Y-Z Ma, et al, (2005) Important Physical Effects: - band structure (~ ev shift each) - quasiparticle self-energy - excitonic Transport gap optical gap Spataru, Ismail-Beigi, Benedict & Louie, PRL (2004)

31 Optical Spectrum of Semiconducting Carbon SWNTs Spataru, Ismail-Beigi, Benedict & Louie (2004) (7,0) (8,0) (10,0) (11,0) Excitonic effects are equally dominant in BN nanotubes and Si nanowires!

32 Absorption Spectrum of (3,3) Metallic Carbon Nanotube E F Existence of bound excitons in metal tubes! (E b = 86 mev) Due to ineffective screening in 1D and symmetric gap Similar results for the (10,10) and larger metallic tubes Spataru, Ismail-Beigi, Benedict & Louie, PRL (2004)

33 Optical Absorption Spectra of Two Metallic SWCNTs Excitons in Metallic CNTs One bright exciton per van Hove singularity (10,10) Exciton binding energy E b ~ mev E b weakly dependent on tube diameter (12,0) Exciton binding enegies E 11 b E 22 b (5,5) (10,10) 60 mev 50 mev 70 mev 60 mev (12,0) 50 mev 120 mev Deslippe, Spataru, Prendergast & Louie, Nanoletters (2007)

34 (10,10) Metallic SWCNT Peak Shape Comparison (broadened with linewidth of 80 mev) Note: black curve is shifted by 50 mev to align with red curve. Interband transitions model With bound exciton (present theory) Energy (ev) Significant optical line-shape difference should be observable Deslippe, Spataru, Prendergast & Louie, Nanoletters (2007)

35 Experimental Absorption Spectrum of Single Suspended (21,21) Metallic SWCNT F. Wang, et al (2007)

36 Exciton in (21,21) Metallic SWNCT: Theory vs. Experiment Free electron-hole interband transition picture

37 Exciton in (21,21) Metallic SWNCT: Theory vs. Experiment Exciton Theory E b = 50 mev R ex = 3.1 nm Theory Experiment [Additional evidence seen in fieldenhanced photocurrent measurements, Mohite, et al (2007)] (Note: 80 mev broadening used in theory) Wang, et al, to be pubished (2007)

38 Science 299,1874 (2003) Hydrogen terminated Si nanowires 1.3 nm 2 nm 2.5 nm 3 nm 5 nm 7 nm STM measurement of SiNW on graphite

39 Optical Spectrum of d=1.2 nm Si Nanowire Optical absorption in Si wire Absorption = 3.2 ev (~3.4 ev expt.) Exciton binding energy > 1 ev! Yang, Spataru, Louie & Chou (2006)

40 Graphene Nanoribbons Phenomenon of electric field-induced halfmetallicity Tunable spin carriers of one type (100% spin polarization) Could be useful for nanoscale spin generation and injection Optical response is also dominated by excitons Son, Cohen and Louie, Nature (2006) Son, Cohen and Louie, PRL (2006) Yang, Son, Cohen and Louie, (2007)

41 Graphene Electronic Structure E Energy E F k x ' unoccupied k y ' occupied r E =hv F k k y k x E 2 = p 2 c 2 2D massless Dirac fermion system

42 Graphene Nanoribbons with Homogenous Edges & Passivated -bonds Armchair Graphene Nanoribbons (N-AGNRs) Simple tight-binding: Metal: N a = 3p+2 Semiconductor: N a = 3p or 3p+1 Zigzag Graphene Nanoribbons (N-ZGNRs) Simple tight-binding: Always metal Ab initio calculations predicted all GNRs have gaps! Son, Cohen and Louie, PRL (2006)

43 Quasiparticle Band Structure and Optical Spectrum of 10-AGNR Armchair-edge nanoribbon Width of w ~ 1.1nm Large exciton binding energy of E b ~1.3 ev Similar strong exciton effects in other nanoribbons Yang, Park, Cohen and Louie (2007)

44 Forces in the Photo-Excited State: Self-trapped Exciton

45 Forces in Excited State For many systems, photo-induced structural changes are important differences between absorption and luminescence self-trapped excitons molecular/defect conformation changes photo-induced desorption Need excited-state forces structural relaxation luminescence study molecular dynamics, etc. GW+BSE approach gives accurate forces in photoexcited state Ismail-Beigi & Louie, Phys. Rev. Lett. 90, (2003)

46 Excited-state Forces E S = E 0 + S R E S = R E 0 + R S E 0 & R E 0 : DFT S : GW+BSE Ismail-Beigi & Louie, Phys. Rev. Lett. 90, (2003).

47 Verification on molecules Excited-state force methodology Proof of principle: tests on molecules - CO, NH 3, GW-BSE force method works well Forces allow us to efficiently find excited-state energy minima Ismail-Beigi & Louie, Phys. Rev. Lett. 90, (2003).

48 SiO 2 ( -quartz): optical properties Emission at ~ 3 ev! [1] Silicon Oxygen [1] Ismail-Beigi & Louie (2004) [2] Philipp, Sol. State. Comm. 4 (1966)

49 Self-trapped exciton (STE) in SiO 2 ( -quartz) Triplet STE has 1 ms and ~ 6 ev Stokes shift [1] 1. Start with 18 atom bulk cell 2. Randomly displace atoms by ±0.02 Å 3. Relax triplet exciton state 4. Repeat steps 2&3: same final config. [1] e.g. Itoh, Tanimura, & Itoh, J. Phys. C 21 (1988). Ismail-Beigi & Louie, PRL (2005)

50 Structural Distortion from Self-Trapped Exciton in SiO 2 Silicon Oxygen Final configuration: Broken Si-O bond Hole on oxygen Electron on silicon Si in planar sp 2 configuration Ismail-Beigi & Louie, PRL (2005)

51 No activation barrier! Atomic rearrangement for STE

52 Electron-Hole Wavefunction of Self-Trapped Exciton in SiO 2 Hole probability distribution with electron any where in the crystal Electron probability distribution given the hole is in the colored box

53 Electron & Hole Distributions of Self-Trapped Exciton in SiO 2 Silicon Oxygen Final configuration: Broken Si-O bond Hole on oxygen (brown) Electron on silicon (green) Si in planar sp 2 configuration Ismail-Beigi & Louie, PRL (2005)

54 Constrained DFT Calculations Constrained LSDA: DFT with excited occupations Problems: Relaxes back to ideal bulk from random initial displacements: excited-state energy surface incorrectly has a barrier. Large initial distortions needed for STE [1,2] Predicted Stokes shift and STE luminescence energy are very poor to correlate with experiments [1] Song et al., Nucl. Instr. Meth. Phys. Res. B , 451 (2000). [2] Van Ginhoven and Jonsson, J. Chem. Phys. 118, 6582 (2003).

55 STE in SiO 2 : Comparison to Experiment Luminescence freq.: T (ev) Stokes shift (ev) Luminescence Pol z (*) Expt. [1-6] 2.6, 2.74, 2.75, , 0.65, 0.70 GW+BSE CLSDA (forced) Tanimura et al., Phys. Rev. Lett. 51, 423 (1983). 2. Tanimura et al., Phys. Rev. B 34, 2933 (1986). 3. Itoh et al., J. Phys. C 21, 4693 (1988). 4. Itoh et al., Phys. Rev. B 39, (1989). 5. Joosen et al., Appl. Phys. Lett. 61, 2260 (1992). 6. Kalceff & Phillips, Phys. Rev. B 52, 3122 (1996). (*) Pol = I z I xy I z + I xy

56 Rohlfing & Louie, PRL, 1998.

57 Molecular energy levels at metal-organic interfaces Metal-organic contacts Energy level diagram Fermi Energy E vacuum LUMO Affinity Level Metal HOMO Ionization Level z Ubiquitous in nanoscale devices Single-molecule junctions, organic electronics, passivators for nanoparticle surfaces, etc Physical Frontier effects molecular orbital alignment? Charge HOMO-LUMO transfer (interface gap? dipoles) Quantum Implications mechanical for charge (electronic) transport? coupling Surface polarization

58 Gas-phase benzene: HOMO-LUMO gap Frontier Levels in the gas phase LDA GW LDA underestimates the gap by a factor of ev 10.5 ev GW HOMO-LUMO gap agrees with experiment (IP-EA) LUMO predicted to be above the vacuum level in GW, in agreement with experiment Experiment: IP EA = ev Neaton, Hybertsen, Louie, PRL (2006)

59 Excited electronic states at the organic-inorganic interface graphite: Frontier electronic orbitals graphite: Energy level diagram Graphite E F 7.3 ev 10.5 ev (5.05) (5.16) Metal-molecule interface Isolated molecule HOMO-LUMO gaps of aromatic molecules are reduced at metal contacts Neaton, Hybertsen, Louie, Phys. Rev. Lett. (2006) Nonlocal electronic correlations between the molecule and substrate are responsible

60 LDA+U as a starting mean-field H for GW quasiparticle calculations - bcc hydrogen - ZnS

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62 H = H o + (H - H o ) Hybertsen and Louie (1985)

63 bcc Hydrogen Energy gap at r s = 4 E QP GGA K-S gap GW gap

64 bcc Hydrogen Energy gap at r s = 4

65 bcc Hydrogen Energy gap at r s = 4

66 bcc Hydrogen energy gap vs. r s

67 bcc Hydrogen energy gap vs. r s Expt: 12.8eV r s = 2.17 (r s VQMC = 2.1 ± 0.1) Kioupakis, Zhang, & Louie (2006)

68 Energy states of ZnS E g expt U = 8 ev J = 1 ev d states (expt) LDA GW (LDA) LDA+U GW (LDA+U) Zhang, Miyake, Cohen and Louie (2006)

69 Summary The GW-BSE approach is a powerful method for studying quasiparticle excitations and photo-excited states of condensed matter. Very robust for a number of moderately correlated systems crystals, surfaces, polymers, nanostructures Present methods can handle up to ~ atoms per supercell. Need improvements to address larger and more correlated systems.

70 Collaborators Mark Hybertsen Eric Shirley John Northrup Michael Rohlfing Eric Chang Sohrab Ismail-Beigi Catalin Spataru Jack Deslippe David Prendergast Li Yang Mei-Yin Chou Young-Woo Son Marvin Cohen Jeff Neaton Manos Kioupakis Peihong Zhang Takashi Miayake