Download CECAM Meeting “Development of Methods for

CECAM Meeting “Development of Methods for Quantum Dynamics in Condensed Phase”, September 16-18, Lyon, France.
Interfacial Electron Transfer and
Quantum Entanglement in Functionalized
TiO2 Nanostructures
Sabas Abuabara, Luis G.C. Rego* and Victor S. Batista
Department of Chemistry, Yale University, New Haven, CT 06520-8107
*Current Address:
Physics Department, Universidade Federal do Parana, CP 19044, Curitiba, PR, Brazil, 81531-990
Aspects of Study
• Interfacial Electron Transfer Dynamics
– Relevant timescales and mechanisms
– Dependence of electronic dynamics on the crystal
symmetry and dynamics
• Effect of nuclear dynamics
– Whether nuclear motion affects transfer mechanism
or timescale
– Implications for quantum coherences
• Hole Relaxation Dynamics
– Decoherence timescale
– Possibility of coherent control
L.G.C. Rego and V.S. Batista, J. Am. Chem. Soc. 125, 7989 (2003)
V.S. Batista and P. Brumer, Phys. Rev. Lett. 89, 5889 (2003), ibid. 89, 28089 (2003)
Highlights of Presentation
Unit Cell for ab initio DFT MD simulations
Electronic Hamiltonian and Propagation Scheme
Electron Injection at 0 K
Electron Injection at 100 K
Hole Dynamics at 100 K
Coherent Control
Model System – Unit Cell
VASP/VAMP simulation package
Hartree and Exchange Correlation Interactions: Perdew-Wang functional
Ion-Ion interactions: ultrasoft Vanderbilt pseudopotentials
TiO2-anatase
nanostructure
functionalized by
an adsorbed
catechol molecule
124 atoms:
32 [TiO2] units = 96
catechol [C6H6-202] unit = 12
16 capping H atoms = 16
Phonon Spectral Density
O-H stretch, 3700 cm-1
(H capping atoms)
C-C,C=C stretch
C-H stretch
1000 cm-1,1200 cm-1
3100 cm-1
TiO2 normal modes
262-876 cm-1
Comments on MD Simulations
Classical MD for nuclei with QM electrons
• state-of-the-art large scale ab initio MD simulation
calculation using IBM SP2 Supercomputer
• relaxed equilibrium structure for T = 0 K Quantum
dynamics
• nuclear trajectories for T = 100 K Quantum dynamics
Highlights of Presentation
Unit Cell for ab initio DFT MD simulations
Electronic Hamiltonian and Propagation Scheme
Electron Injection at 0 K
Electron Injection at 100 K
Hole Dynamics at 100 K
Coherent Control
Simulations of Electronic Relaxation
Accurate description of charge delocalization requires
simulations in extended model systems.
•Simulations in small clusters (e.g., 1.2 nm nanostructures) are affected
by surface states that speed up the electron injection process
• Periodic
boundary conditions alone often
recurrencies (back-electron transfer events).
artificial
Three unit cells
extending the
system in
[-101] direction
[-101]
System
extended in
the [010]
direction
introduce
[010]
Electronic Hamiltonian
..
H is the Extended Huckel Hamiltonian in the basis of
Slater type atomic orbitals (AO’s) including
• 4s, 3p and 3d AO’s of Ti 4+ ions
• 2s and 2p AO’s of O 2- ions
• 2s and 2p AO’s of C atoms
• 1s AO’s of H atoms
• 596 basis functions per unit cell
S is the overlap matrix in the AO’s basis set.
How good is this tight binding Hamiltonian?
Electronic Density of States (1.2 nm particles)
photoexcitation
LUMO,LUMO+1
HOMO
HOMO
Valence Band
ZINDO1 Band gap =3.7 eV
Exp. (2.4 nm) = 3.4 eV
Exp. (Bulk-anatase) = 3.2 eV
Band gap =3.7 eV
Conduction Band
Comments on Propagation Scheme
Unlike gas phase MD,
condensed phase has many
bound states
Born-Oppenheimer Potential
Energy Surfaces are
approximately parallel so
that equilibrium nuclear
dynamics is valid
Underlying ‘simplicity’ of system allows use of ‘easy’ procedure
to simulate quantum dynamics of complex, extended System
Mixed Quantum-Classical Dynamics
Propagation Scheme
ˆ (t )  e
(t )  Uˆ (t ) (0) , where U
and
(t )   Bq (t ) q(t )
i


 H ( t ') dt '
with
q
q(t )   Ci ,q (t ) Ki (t )
are the instantaneous MO’s
i
obtained by solving the extended-Hückel generalized
eigenvalue equation:
H (t )C (t )  S (t )C (t ) E (t )
Propagation Scheme cont’d
Derive propagator for midpoint scheme:
Hamiltonian changes linearly during time step /
Forward and Backwards propagation equal
Uˆ (t   2 )  (t )   Bq (t )e
i

 Eq ( t )

2
q(t )
q
Uˆ (t   2 , t   )  (t   ) 
 B (t   )e
q
q
i

Eq ( t  )

2
q (t   )
Propagation Scheme cont’d
=
Set
and multiply by MO at iterated time:
Bq ( t   )   B p ( t ) e
i

 ( E p ( t )  Eq ( t  ))

2
q( t   ) p ( t )
p
Which in

0 limit we approximate as
Bq (t   )  Bq (t )e
i

 ( E p ( t )  Eq ( t  ))

2
Propagation Scheme cont’d
With this scheme, we can calculate for all t>0 :
• electronic wavefunction
• electronic density
• Define the Survival Probability for electron
to be found on initially populated adsorbate
molecule
PMOL (t ) 
SYS MOL
  C
*
i ,
j,
i,
i, j
 ,
(t )C j ,  (t )S
Highlights of Presentation
Unit Cell for ab initio DFT MD simulations
Electronic Hamiltonian and Propagation Scheme
Electron Injection at 0 K
Electron Injection at 100 K
Hole Dynamics at 100 K
Coherent Control
Injection from LUMO (frozen lattice, 0 K)
TiO2 system
extended in [-101]
direction with PBC
in [010] direction
Grey ‘Balloons’
are isosurface of
electronic density
(not integral!)
Good Picture of
MO
Allow visualization
of mechanism
LUMO Injection (frozen lattice) cont’d
Note effect of nodal
plane in density near
Ti4+ ions anchoring
adsorbate
LUMO Injection (frozen lattice)
Injection from LUMO+1 (frozen lattice, 0 K)
Note different
symmetry for
LUMO+1
nodal plane in
density near Ti4+
under adsorbate
LUMO+1 Injection (frozen lattice)
Comments on Frozen Lattice Results
• Description of charge delocalization requires simulations in
extended model systems. Simulations in smaller clusters are
affected by surface states that speed up the electron injection
process, and periodic boundary conditions often introduce
artificial recurrencies.
• Reaction mechanisms and characteristic times for electron
injection in catechol/TiO2-anatase nanostructure are highly
sensitive to the symmetry of the initially populated electronic
state.
• Electron Injection from catechol LUMO involves a primary step
within 5 fs localizing injected charge on the dxz orbital of the
penta-coordinated Ti4+ ion next to the adsorbate (coordination
complex ligand mechanism).
Comments on Frozen Lattice Results
•primary event is followed by charge delocalization (i.e., carrier relaxation)
through the anatase crystal. At low temperature, this is an anisotropic
process that involves surface charge separation along the [101] direction of
the anatase crystal. Carrier relaxation along the [-101] direction can be much
slower than along the [101] and [010] directions.
•in contrast to the LUMO relaxation, electron injection from the catechol(LUMO+1) involves coupling to the dxz orbitals of the Ti4+ ions directly
anchoring the adsorbate. Here, both the primary and secondary steps are
faster than electron injection from LUMO. Also, in contrast to injection from
LUMO, the charge delocalization process involves charge diffusion along the
semiconductor surface (i.e., along the [010] direction in the anatase crystal)
before the injected charge separates from the surface by diffusion along the
[101] direction.
Highlights of Presentation
Unit Cell for ab initio DFT MD simulations
Electronic Hamiltonian and Propagation Scheme
Electron Injection at 0 K
Electron Injection at 100 K
Hole Dynamics at 100 K
Coherent Control
Injection from LUMO (T = 100 K Lattice)
TiO2 system
extended in [-101]
direction with PBC
in [010] direction
Compare
mechanisms
and
track electronic
density,
subtract T = 0 K
balloons from
T = 100 K
Influence of Phonons on Electron Injection cont’d
[-101] system; effect of motion on same initial cond’s
t = 4.8 fs
Surplus
Deficiency
Influence of Phonons on Electron Injection cont’d
[-101] system; effect of motion on same initial cond’s
t = 1.6 fs
Surplus
Deficiency
LUMO Injection at Finite Temperature (100 K)
T = 0K
LUMO+1 Injection at Finite Temperature (100 K)
T = 0K
Comments on Thermal Lattice Results
• We have shown that the anisotropic nature of
carrier relaxation as well as the overall injection
process are significantly influenced by temperature
• electron-phonon scattering induces transient
couplings from the AOs of those Ti4+ atoms critical to
the electron transfer mechanism to delocalized
electronic states within the semiconductor
• electron-phonon scattering also induces ultrafast
electron transfer along the mono-layer of adsorbate
molecules.
Highlights of Presentation
Unit Cell for ab initio DFT MD simulations
Electronic Hamiltonian and Propagation Scheme
Electron Injection at 0 K
Electron Injection at 100 K
Hole Dynamics at 100 K
Coherent Control
Relaxation Dynamics of Hole States Localized on
Adsorbate Monolayer
After photoinduced
electron-hole pair
separation,
(electron excited to
LUMO/+1 and injects)
Compute Hole Population on
each adsorbate
PMOL (t ) 
SYS MOL
  C  (t )C
*
i,
j,
j ,
(t ) Si ,,j
i,
Hole is left behind, off
resonant w.r.t. conduction
and valence bands
Dynamics on Adsorbate
Monolayer
t=15 ps
Super-exchange hole transfer
Coherent Hole-Tunneling Dynamics
SURVIVAL PROBABILITY
1.0
0.8
0.6
0.4
0.2
0.0
0
20
40
60
80
100
TIME (PS)
Pj (t ) 

2
2
jk
/

2
sin
2
(  jk t );
2
jk
2
2

(
/
)

(
/
2
)

 jk
  jk
jk
Elements of subspace reduced density matrix
TIME (PS)
Elements
Investigation
of subspace
of Coherences
reduced density
cont’d
matrix
TIME (PS)
Measure of Decoherence / Impurity of Time Evolved Wavefunction
Highlights of Presentation
Unit Cell for ab initio DFT MD simulations
Electronic Hamiltonian and Propagation Scheme
Electron Injection at 0 K
Electron Injection at 100 K
Hole Dynamics at 100 K
Coherent Control
Investigation of Coherent-Control
If hole wavefunction is mostly pure or, conversely,
If initial wavefunction has not completely decohered…
One can manipulate the underlying quantum dynamics by merely
affecting the phase of the state, using femtosecond laser pulses
CB
C
L
12
superexchange
VB
TiO2
semiconductor
Adsorbate molecules (C, L,…)
Investigation of Coherent-Control cont’d
Apply pulsed radiation tuned to perturbed transition frequency 21
e.g., 2-p pulses (200 fs spacing)
Agarwal et. al. Phys. Rev. Lett. 86, 4271 (2001)
Results in unitary operation
 (t )
 = 200 fs, 12


t= k*
 (t )  2
 ( 0)  ( t )
 ( 0)  ( 0)
 ( 0)
Keeping adsorbate populations constant by destroying phase
relations between them, disallowing interference.
Investigation of Coherent-Control cont’d
SURVIVAL PROBABILITY
1.0
2-p pulses (200 fs spacing)
60 fs
14 fs
0.8
0.6
0.4
0.2
0.0
0
20
40
60
TIME (PS)
80
100
Investigation of Coherent-Control cont’d
SURVIVAL PROBABILITY
1.0
2-p pulses (200 fs spacing)
0.8
2 fs
42 fs
0.6
0.4
0.2
0.0
0
20
40
60
TIME (PS)
80
100
Comments on Hole Relaxation Dynamics
• We have investigated the feasibility of creating
entangled hole-states localized deep in the
semiconductor band gap.
•These states are generated by electron-hole pair
separation after photo-excitation of molecular surface
complexes under cryogenic and vacuum conditions.
•These states persist despite the decohering action
of thermal nuclear motion
Acknowledgment
•NSF Nanoscale Exploratory Research (NER) Award ECS#0404191
•NSF Career Award CHE#0345984
•ACS PRF#37789-G6
•Research Corporation, Innovation Award
•Hellman Family Fellowship
•Anderson Fellowship
•Yale University, Start-Up Package
•NERSC Allocation of Supercomputer Time
•ALL OF OUR HOSTS esp.
CECAM!
Thank you !