Excitation Dynamics in Quantum Dots. Oleg Prezhdo U. Washington, Seattle

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1 Excitation Dynamics in Quantum Dots Oleg Prezhdo U. Washington, Seattle Warwick August 27, 2009

2 Outline Time-Domain Density Functional Theory & Nonadiabatic Molecular Dynamics Quantum backreaction, surface hopping Zero-point energy Decoherence/dephasing Excitation Dynamics in Quantum Dots Phonon bottleneck Generation of multiple excitons Dephasing, linewidths, biexciton fission

3 Adiabatic vs. Nonadiabatic MD Nonadiabatic MD: Coupling between potential surfaces opens channels for system to change electronic states. transition allowed electrons treated quantum-mechanically e e e e e e nuclei treated classically weak coupling strong coupling

4 Time-Domain DFT for Nonadiabatic Molecular Dynamics ( ) = p p x x 2 ) ( ϕ ρ ( ) ( ) ( ) SD N v q p t x t x t x,, 2, 1 ϕ ϕ ϕ K = Ψ Electron density derives from Kohn-Sham orbitals H ( ) t R DFT functional depends on nuclear evolution ( ) ( ) t x H t t x i p p,, ϕ ϕ = h K = 1,2 p Variational principle gives = R i c c i R β α αβ β β β α χ χ δ ε h h Orbitals are expanded in adiabatic KS basis ( ) ( ) ( ) = α α α χ ϕ x t c t x p p, ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) t R x t R t R x t R x H ; ; ; α α α χ χ = ε

5 Open Theoretical Questions How to couple quantum and classical dynamics? quantum influence on classical trajectory Can one do better than classical mechanics for nuclear motion? zero-point motion, tunneling, branching, loss of coherence

6 Nuclear Evolution: Ehrenfest Total energy of electrons and nuclei E M R tot = + V R t + Trx ρ x H x; 2 de tot is conserved 2 dt ( ()) ( ) ( R( t) ) = 0 time-dependent Hellmann-Feynman theorem gives Newton equation M R quantum force ( x) R H ( x R( t) ) = R V Tr ρ ; x

7 Nuclear Evolution: Surface Hopping a.k.a., quantum-master equation with time-dependent transition rates: - non-perturbative - correct short time dynamics Trajectory branching: Tully, JCP 93, 1061 (1990); Within TDDFT: Craig, Duncan, Prezhdo PRL 95, (2005) Detailed balance: Parahdekar, Tully JCP 122, (2005)

8 Quantum-Classical Lie Bracket O. V. Prezhdo, V. V. Kisil Phys. Rev. A (1997) O. V. Prezhdo J. Chem. Phys (2006) quantum commutator + classical Poisson bracket problems with Jacobi identity:

9 Quantized Hamilton Dynamics O. V. Prezhdo, Yu. V. Pereverzev J. Chem. Phys. 113, 6557 (2000) O. V. Prezhdo Theor. Chem. Acc. 116, 206 (2006) V = q2 2 + q3 3 d < q > dt = < p >; d < p > dt =- <q >- <q 2 > but < q 2 > < q > < q > and d < q 2 > dt = < pq + qp > 2 < pq > s d < pq > s dt = < p 2 >- <q 2 >- <q 3 > the infinite hierarchy is terminated by a closure < q 3 > 3 < q 2 > < q > 2 < q > 3

10 Harmonic Oscillator in Mapped QHD-2 hbar mass hbar mass

11 Metastable Cubic Potential in Mapped QHD-2 hbar mass hbar mass

12 Double-Slit Potential in Mapped QHD-2 potential seen by a narrow wavepacket potential seen by a wide wavepacket V HqL VH2L HqL s 2

13 Schrodinger Cat and Decoherence B > 0 atom alive cat System - radioactive atom; Bath - cat dead In Nanomaterials System - electrons, spins; Bath - phonons

14 Franck-Condon Factor and Decoherence B > 0 ΣB 2 B 1 B 2 2 δ E1 E 2 = e ie 1 E 2 t/h B 1 t B 2 t dt Bath (vibrational) wave functions diverge This affects evolution of (electronic) system

15 Decoherence and Surface Hopping Reduced density matrix: ρ 11 ρ 12 ρ 21 ρ 22 With decoherence: Without decoherence ρ 11 ρ 12 B 2 B 1 ρ = B ρ S B B ρ 12 0 on decoherence time scale hopping probability P 12~ρ 12 ρ 21 B 1 B 2 ρ 22 2 P 12 = T 12 + T P 12 = T 12 + T Decoherence makes transitions less likely < (quantum Zeno effect)

18 Quantum Dot Solar Cells

19 Open Questions in Nanoscale Materials Quantum confinement effects on excitation dynamics: To what extent are there bottlenecks? Electron-vibrational relaxation (heating): Which phonons are involved and why?

21 Biexcitons in PbSe Quantum Dots Shaller, Klimov PRL (2004); Ellingson, Beard, Johnson, Yu, Micic, Nozik, Shabaev, Efros, NanoLett (2005) Biexciton creation yield Exciton to biexciton time under 0.25ps

Electron-Phonon Relaxation in PbSe Quantum Dots Schaller, Pietryga, Goupalov, Petruska, Ivanov, Klimov PRL 95

22 Electron-Phonon Relaxation in PbSe Quantum Dots Schaller, Pietryga, Goupalov, Petruska, Ivanov, Klimov PRL (2005) No phonon bottleneck. Times are similar to biexciton creation times Larger dots relax more slowly?!

23 Structural relaxation of PbSe Quantum Dots 32 atoms Pb 16 Se 16 d=0.9nm 136 atoms Pb 68 Se 68 d=1.3nm Bulk, T=0 K Relaxed, T=0 K Heated, T=300 K Bulk, T=0 K Relaxed, T=0 K Heated, T=300 K Even a very small PbSe quantum dot preserves its bulk topology

24 Orbitals and Density of States (LUMO) (LUMO+1) (LUMO+2) (LUMO+3) (LUMO) (LUMO+1) (LUMO+2) (LUMO+3) DOS of Pb 16 Se 16 Time, ps states mix; sp 3 hybridization (?) DOS of Pb 68 Se hole states electronic states E-E f, ev

25 Absorption Spectra Pb 16 Se 16 oscillator strength Pb 68 Se 68 oscillator strength Energy, ev Energy, ev

26 Time, fs Comparison of Relaxation Population of 4 states near to threshold hole states (VB) electron states (CB) Times agree with experiment Similar relaxation times for electrons and holes Biexciton creation is faster than relaxation Larger dot relaxes more slowly due to weaker NA coupling

27 Active Phonon Modes Electron-Phonon Coupling: Lower frequency acoustic modes are more active than optical modes Raman Spectrum of 3-nm PbS QD [Acc Chem. Res. 2000, 33, ]

Science 322 929 (2008) Many factors must

28 Phonon Bottleneck for 1P Electron in CdSe Quantum Dots Pandey, Guyot-Sionnest, Science (2008) Many factors must be avoided. 1P to 1S electron relaxes within ~1ns

29 Phonon Bottleneck for 1P Electron in Cd Se Quantum Dots big 1P 1S electron gap ZnS shell does not change electronic structure of CdSe core

30 Geometric and Electronic Structure of Ge and Si Clusters DOS is symmetric, slightly higher for electrons

31 Relaxation in Ge and Si Clusters Electrons relax much faster than holes! (despite nearly symmetric DOS)

32 Active Phonon Modes Low frequency modes are active for both electrons and holes However, high frequency modes are active only for electrons

33 Proposed Multiplication Mechanisms Inverse Auger Dephasing Direct Excitation

34 Hartree-Fock Band Structure 1. Small dots represent large dot DOS 2. Huge one-electron gap 3. Symmetric vs. asymmetric DOS 4. Secondary gaps in PbSe DOS

35 SAC-CI Spectra and Fraction of Multiple Excitons spectra fraction of multiple excitons PbSe CdSe CdSe spectra agree with experiment JACS 128, 629 (2006) 1. Sharp onset of multiple excitons 2. Above threshold: double excitons in PbSe; single, double and superpositions in CdSe

36 New Experimental Data on Multiple Exciton Generation Ideal Klimov et al. Bawendi et al. Bulk Apparent increase in Static decay due to ionization, etc.

37 Calculations for Charged PbSe Dots Conduction band transitions overwhelm MEs Much higher ME threshold

38 Phonon-Induced Pure-Dephasing Times, fs 300K/100K ME Fission much slower Smaller dots faster dephasing Lower Temperature slower dephasing

Summary for Quantum Dots No bottleneck due to small gaps Bottleneck only at lowest energy PbSe Smaller dots relax faster, coupling over DOS Acoustic, not optical modes are active All

39 Summary for Quantum Dots No bottleneck due to small gaps Bottleneck only at lowest energy PbSe Smaller dots relax faster, coupling over DOS Acoustic, not optical modes are active All three MEG mechanisms are important: inverse Auger, dephasing and direct (PbSe) CdSe/ZnS Charged QDs show much lower ME yields Surface ligands are important Dephasing: 10fs lumin, MEG, 100fs MEF

41 General Questions Quantum confinement effects on excitation dynamics: To what extent are there bottlenecks? Electron-vibrational relaxation (heating): Which phonons are involved and why? ih χ α R χ β R

42 Dots Nano Lett (2006) Nano Lett (2007) J.Phys.Chem.C (2007) J.Phys.Chem.C (2008) J. Phys.Chem.C (2008) Chem.Phys. Lett (2008) J.Photochem.-Photobiol.A 190, 342 (2008) Pure&Appl.Chem (2008) Chem.Phys.Lett. FRONTIER (2008) ACS-Nano 3 93 (2009) Phys. Rev. B (2009) J. Phys. Chem. C (2009) Dalton Trans. in press ACS-Nano in press Acc. Chem. Research, submitted NAMD/TDDFT TDDFT: Phys. Rev. Lett (2005) Bracket: Phys. Rev. A 56, 162 (1997) J. Chem. Phys. Rapid. 124, (2006) Decoher: J. Chem. Phys (1999) Phys. Rev. Lett. 85, 4413 (2000) Bohmian: Phys. Rev. Lett (2001) Rev. Comp. Chem. in press QHD: J. Chem. Phys. 113, 6557 (2000); 116, 4450 (2002); 116, 8704 (2002); 117, 2995 (2002); (2004); (2004); (2005); (2007); 129, (2008) Chem. Phys. Lett. 346, 463 (2001); 378, 533(2003) J. Mol. Struct (2003) Adv.Top.Theor.Chem.Phys. 12B 339 (2003) Theor. Chem. Acc. 116, 206 (2006) J. Phys. Chem. A 111, (2007) J. Phys. Soc. Jap (2008)

Nonadiabatic Molecular Dynamics with Kohn-Sham DFT: Modeling Nanoscale Materials. Oleg Prezhdo

Nonadiabatic Molecular Dynamics with Kohn-Sham DFT: Modeling Nanoscale Materials Oleg Prezhdo CSCAMM Mar 11, 2010 Outline Nonadiabatic MD with Kohn-Sham DFT Advantages & Validity Quantum Backreaction &