*current address: physics department, universidade federal do parana, cp 19044, curitiba, pr,...

*Current Address:

Physics Department, Universidade Federal do Parana, CP 19044, Curitiba, PR, Brazil, 81531-990

Sabas Abuabara, Luis G.C. Rego* and Victor S. BatistaDepartment of Chemistry, Yale University, New Haven, CT

06520-8107

Interfacial Electron Transfer and Quantum Entanglement in

Functionalized TiO2 Nanostructures

CECAM Meeting “Development of Methods for Quantum Dynamics in Condensed Phase”, September 16-18, Lyon, France.

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 100 K


Coherent Control

Hole Dynamics at 100 K

Model System – Unit Cell

TiO2-anatase nanostructure functionalized by an adsorbed catechol molecule

124 atoms:

32 [TiO2] units = 96catechol [C6H6-202] unit = 1216 capping H atoms = 16

VASP/VAMP simulation packageHartree and Exchange Correlation Interactions: Perdew-Wang

functional Ion-Ion interactions: ultrasoft Vanderbilt pseudopotentials

Phonon Spectral Density

O-H stretch, 3700 cm-1

(H capping atoms)

C-H stretch

3100 cm-1

TiO2 normal modes

262-876 cm-1

C-C,C=C stretch

1000 cm-1,1200 cm-1

Comments on MD SimulationsClassical 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


Unit Cell for ab initio DFT MD simulationsElectronic Hamiltonian and Propagation

Scheme



Coherent Control


System extended in the [010] direction

[010]

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 introduce artificial recurrencies (back-electron transfer events).

Simulations of Electronic Relaxation

Three unit cells extending the system in [-101] direction

[-101]

Electronic HamiltonianH 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?

HOMO

LUMO,LUMO+1

Valence Band Conduction BandBand gap =3.7 eV

ZINDO1 Band gap =3.7 eV

Exp. (2.4 nm) = 3.4 eV

Exp. (Bulk-anatase) = 3.2 eV

Electronic Density of States (1.2 nm particles)

HOMO

photoexcitation

Comments on Propagation Scheme

Underlying ‘simplicity’ of system allows use of ‘easy’ procedure to simulate quantum dynamics of complex, extended System

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

Mixed Quantum-Classical Dynamics Propagation Scheme

)0()(ˆ)( tUt

and withq

qq ttBt )()()(

')'(

)(ˆ dttHi

etU , where

i

iqiq tKtCt )()()( , are the instantaneous MO’s

)()()()()( tEtCtStCtH

obtained by solving the extended-Hückel generalized eigenvalue equation:

Propagation Scheme cont’d

Derive propagator for midpoint scheme:Hamiltonian changes linearly during time step /

Forward and Backwards propagation equal

)()()()(ˆ 2)(

2 tqetBttUq

tEi

q

q

)()(

)(),(ˆ

2)(

2

tqetB

tttU

tEi

qq

q

Set = and multiply by MO at iterated time:

)()()()( 2))()((

tptqetBtBp

tEtEi

pq

qp

2))()((

)()(

tEtE

i

qq

qp

etBtB

Which in 0 limit we approximate as


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

SYS

j

jij

MOL

iiMOL StCtCtP

,

,,,

,

*, )()()(





Electron Injection at 0 KElectron Injection at 100 K

Coherent Control


Injection from LUMO (frozen lattice, 0 K)

TiO2 system extended in [-101] direction with PBC in [010] directionGrey ‘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) P

MO

L(t)

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)

PM

OL(

t)

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.






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

t = 4.8 fs

Influence of Phonons on Electron Injection cont’d [-101] system; effect of motion on same initial cond’s

SurplusSurplus DeficiencyDeficiency

t = 1.6 fs

Influence of Phonons on Electron Injection cont’d [-101] system; effect of motion on same initial cond’s

SurplusSurplus DeficiencyDeficiency

LUMO Injection at Finite Temperature (100 K)

T = 0K

LUMO+1 Injection at Finite Temperature (100 K)

T = 0K

• 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.

Comments on Thermal Lattice Results






Coherent Control


Relaxation Dynamics of Hole States Localized on Adsorbate Monolayer

t=15 ps Super-exchange hole transfer

SYS

j

jij

MOL

iiMOL StCtCtP

,

,,,

,

*, )()()(

Compute Hole Population on each adsorbateAfter photoinduced

electron-hole pair separation,

(electron excited to LUMO/+1 and injects)

Hole is left behind, off resonant w.r.t.

conduction and valence bands

Dynamics on Adsorbate Monolayer

0 20 40 60 80 1000.0

0.2

0.4

0.6

0.8

1.0

SU

RV

IVA

L P

RO

BA

BIL

ITY

TIME (PS)

Coherent Hole-Tunneling Dynamics

);(/

)(22

2

22

sin ttPjk

jk

jkj

22 )2/()/( jkjkjk

PM

OL(

t)P

MO

L(t)

Elements of subspace reduced density matrix

TIME (PS)

Investigation of Coherences cont’d

TIME (PS)

Elements of subspace reduced density matrix

Measure of Decoherence / Impurity of Time Evolved Wavefunction


Coherent Control






Investigation of Coherent-Control

If hole wavefunction is mostly pure or, conversely,If initial wavefunction has not completely decohered…

CB

VB

CL

superexchange

Adsorbate molecules (C, L,…)TiO2 semiconductor

One can manipulate the underlying quantum dynamics by merely affecting the phase of the state, using femtosecond laser pulses

)0()0()0(

)()0(2)()(

t

ttt= k*

= 200 fs,

Investigation of Coherent-Control cont’d

2- pulses (200 fs spacing) Agarwal et. al. Phys. Rev. Lett. 86, 4271

(2001)

Apply pulsed radiation tuned to perturbed transition frequency 21

e.g.,

Results in unitary operation

Keeping adsorbate populations constant by destroying phase relations between them, disallowing interference.


0 20 40 60 80 1000.0

0.2

0.4

0.6

0.8

1.0

S

UR

VIV

AL

PR

OB

AB

ILIT

Y

TIME (PS)

2- pulses (200 fs spacing)

14 fs 60 fs

0 20 40 60 80 1000.0

0.2

0.4

0.6

0.8

1.0

S

UR

VIV

AL

PR

OB

AB

ILIT

Y

TIME (PS)

2- pulses (200 fs spacing)

2 fs 42 fs


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

•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 !

Acknowledgment

*current address: physics department, universidade federal do parana, cp 19044, curitiba, pr,...

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