Excitation Dynamics in
Quantum Dots
Oleg Prezhdo
U. Washington, Seattle
Warwick – August 27, 2009
Outline
¾ Time-Domain Density Functional Theory
& Nonadiabatic Molecular Dynamics
•
•
•
Quantum backreaction, surface hopping
Zero-point energy
Decoherence/dephasing
¾ Excitation Dynamics in Quantum Dots
•
•
•
Phonon bottleneck
Generation of multiple excitons
Dephasing, linewidths, biexciton fission
Adiabatic vs. Nonadiabatic MD
Nonadiabatic MD: Coupling
between potential surfaces
opens channels for system to
change electronic states.
electrons treated quantum-mechanically
e
e
transition allowed
e
e
e
e
nuclei treated classically
weak coupling
strong coupling
Time-Domain DFT for
Nonadiabatic Molecular Dynamics
Electron density derives from Kohn-Sham orbitals
ρ (x ) = ∑ ϕ p ( x )
2
Ψ = ϕ p (x1 , t )ϕ q (x2 , t )Kϕ v (x N , t )
p
SD
DFT functional H depends on nuclear evolution R(t )
Variational principle gives
ih
∂ϕ p ( x, t )
∂t
= Hϕ p ( x, t ) p = 1,2 K
α
α
Orbitals are expanded in adiabatic KS basis ϕ p ( x, t ) = ∑ c p (t )χ ( x )
α
H ( x; R (t ))χ ( x; R (t )) = ε (R (t ))χ ( x; R (t ))
α
α
α
•
⎛
•
→
→⎞
ih cα = ∑ c β ⎜ ε β δ αβ − ih χ α ∇ R χ β ⋅ R ⎟
⎜
⎟
β
⎝
⎠
Open Theoretical Questions
9 How to couple quantum and classical dynamics?
quantum influence on classical trajectory
9 Can one do better than classical mechanics for nuclear motion?
zero-point motion, tunneling, branching, loss of coherence
Nuclear Evolution: Ehrenfest
• 2
MR
Total energy of
E =
+ V (R (t )) + Trx ρ ( x )H ( x; R (t ))
electrons and nuclei tot
2
dEtot
=0
is conserved
dt
time-dependent Hellmann-Feynman theorem gives Newton equation
••
→
→
M R = − ∇ R V − Trx ρ ( x ) ∇ R H ( x; R (t ))
quantum force
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, 163001 (2005)
Detailed balance:
Parahdekar, Tully JCP 122, 094102 (2005)
Quantum-Classical Lie Bracket
O. V. Prezhdo, V. V. Kisil Phys. Rev. A 56 162 (1997)
O. V. Prezhdo J. Chem. Phys. 124 201104 (2006)
quantum commutator + classical Poisson bracket
problems with Jacobi identity:
Quantized Hamilton Dynamics
O. V. Prezhdo, Yu. V. Pereverzev J. Chem. Phys. 113, 6557 (2000)
O. V. Prezhdo Theor. Chem. Acc. 116, 206 (2006)
q2 q3
+
V=
2
3
d<q>
dt
but
d < q2 >
dt
= < p >;
d<p>
dt
= - < q > - < q2 >
< q2 > ≠ < q > < q >
= < pq + qp > ≡ 2 < pq >s
the infinite hierarchy is
terminated by a closure
d < pq >s
dt
and
= < p2 > - < q2 > - < q3 >
< q3 > ≈ 3 < q2 > < q > − 2 < q >3
Harmonic Oscillator
in Mapped QHD-2
hbar
mass
hbar
mass
Metastable Cubic Potential
in Mapped QHD-2
hbar
mass
hbar
mass
Double-Slit Potential
in Mapped QHD-2
potential seen by
a narrow wavepacket
V HqL +
potential seen by
a wide wavepacket
1
2
VH2L HqL s2
Schrodinger Cat and Decoherence
| B0 >
alive
atom
cat
System - radioactive atom; Bath - cat
In Nanomaterials
System - electrons, spins; Bath - phonons
dead
Franck-Condon Factor
and Decoherence
| B0 >
Σ
B2
=
B1 B2
ei E
1
– E 2 t/h
2
δ E1 – E 2
B1 t B2 t
dt
Bath (vibrational) wave
functions diverge
This affects evolution of (electronic) system
Decoherence and Surface Hopping
ρ = B ρ S–B B
Reduced density matrix:
ρ 11 ρ 12
ρ 21 ρ 22 →
ρ 11
ρ 12 B2 B1
ρ 21 B1 B2
ρ 22
ρ12 →0 on decoherence
time scale
hopping probability
2
P12 ~ρ12
2
With decoherence: P 12 = T 12 + T 12 + ...
2
Without decoherence P12 = T 12 + T 12 + ...
Decoherence makes transitions less likely
0.1
2
+ 0.1
2
< 0.1 + 0.1
2
(quantum Zeno effect)
Quantum Dot
Solar Cells
Open Questions in
Nanoscale Materials
9 Quantum confinement effects on excitation dynamics:
To what extent are there bottlenecks?
9 Electron-vibrational relaxation (heating):
Which phonons are involved and why?
Biexcitons in PbSe Quantum Dots
Shaller, Klimov PRL 92 186601 (2004); Ellingson, Beard, Johnson,
Yu, Micic, Nozik, Shabaev, Efros, NanoLett 5 865 (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 196401 (2005)
No phonon bottleneck.
Times are similar to
biexciton creation times
Larger dots relax more slowly ?!
Structural relaxation of
PbSe Quantum Dots
32 atoms Pb16Se16 d=0.9nm
Bulk,
T=0 K
Relaxed,
T=0 K
Heated,
T=300 K
136 atoms Pb68Se68 d=1.3nm
Bulk,
T=0 K
Relaxed,
T=0 K
Heated,
T=300 K
Even a very small PbSe quantum dot
preserves its bulk topology
Orbitals and Density of States
(LUMO)
(LUMO) (LUMO+1) (LUMO+2) (LUMO+3)
(LUMO+1) (LUMO+2) (LUMO+3)
states mix; sp3 hybridization (?)
DOS of
Pb68Se68
Time, ps
DOS of
Pb16Se16
3
2
1
electronic
states
hole states
-2
-1
0
E-Ef, eV
1
2
Absorption Spectra
Pb16Se16
oscillator
strength
Energy, eV
Pb68Se68
oscillator
strength
Energy, eV
Population of 4 states near to
threshold
Comparison of Relaxation
Times agree with experiment
Similar relaxation times
for electrons and holes
hole states (VB)
Biexciton creation is faster
than relaxation
Larger dot relaxes more slowly
electron states (CB) due to weaker NA coupling
Time, fs
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, 773-780]
Phonon Bottleneck for 1P
Electron in CdSe Quantum Dots
Pandey, Guyot-Sionnest, Science 322 929 (2008)
Many factors must be avoided.
1P to 1S electron relaxes within ~1ns
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
Geometric and Electronic
Structure of Ge and Si Clusters
DOS is symmetric,
slightly higher for electrons
Relaxation in Ge and Si Clusters
Electrons relax much faster than holes ! (despite nearly symmetric DOS)
Active Phonon Modes
Low frequency modes
are active for both
electrons and holes
However,
high frequency
modes are active
only for electrons
Proposed Multiplication Mechanisms
Inverse Auger
Dephasing
Direct Excitation
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
SAC-CI Spectra and
Fraction of Multiple Excitons
fraction of multiple excitons
spectra
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
New Experimental Data on
Multiple Exciton Generation
“Ideal”
Klimov et al.
Bawendi et al.
Bulk
Apparent increase in Static
decay due to ionization, etc.
Calculations for Charged PbSe Dots
Conduction band
transitions
overwhelm MEs
Much higher
ME threshold
Phonon-Induced Pure-Dephasing
Times, fs
300K/100K
ME Fission – much slower
Smaller dots – faster dephasing
Lower Temperature – slower dephasing
Summary for Quantum Dots
9 No bottleneck due to small gaps
Bottleneck only at lowest energy
PbSe
9 Smaller dots relax faster, coupling over DOS
9 Acoustic, not optical modes are active
9 All three MEG mechanisms are important:
inverse Auger, dephasing and direct (PbSe)
9 Charged QDs show much lower ME yields
9 Surface ligands are important
9 Dephasing: 10fs lumin, MEG, 100fs MEF
CdSe/ZnS
General Questions
9 Quantum confinement effects on excitation dynamics:
To what extent are there bottlenecks?
•
→
→
9 Electron-vibrational relaxation (heating):
α
ih χ ∇ R χ β ⋅ R
Which phonons are involved and why?
Dots
NAMD/TDDFT
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J. Chem. Phys. Rapid. 124, 201104 (2006)
Nano Lett. 6 2295 (2006)
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Nano Lett. 7 3260 (2007)
Phys. Rev. Lett. 85, 4413 (2000)
J.Phys.Chem.C 111 4871 (2007)
Bohmian: Phys. Rev. Lett. 86 3215 (2001)
J.Phys.Chem.C 112 7800 (2008)
Rev. Comp. Chem. in press
J. Phys.Chem.C 112 18291 (2008)
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Chem.Phys. Lett. 458 113 (2008)
116, 4450 (2002); 116, 8704 (2002);
J.Photochem.-Photobiol.A 190, 342 (2008) 117, 2995 (2002); 120 11209 (2004);
Pure&Appl.Chem. 80 1433 (2008)
121 10967 (2004); 122 234109 (2005);
Chem.Phys.Lett. FRONTIER 460 1 (2008) 126 204108 (2007); 129, 144104 (2008)
ACS-Nano 3 93 (2009)
Chem. Phys. Lett. 346, 463 (2001); 378,
Phys. Rev. B 79 235306 (2009)
533(2003) J. Mol. Struct. 630 45 (2003)
J. Phys. Chem. C 113 12617 (2009)
Adv.Top.Theor.Chem.Phys. 12B 339 (2003)
Dalton Trans. in press
Theor. Chem. Acc. 116, 206 (2006)
ACS-Nano in press
J. Phys. Chem. A 111, 10212 (2007)
Acc. Chem. Research, submitted
J. Phys. Soc. Jap. 77 044001 (2008)