1. Sabas Abuabara, Luis G.C. Rego and Victor S. Batista
CNLS Workshop Quantum and Semiclassical Molecular Dynamics of Nanostructures July 15-17, Los Alamos, NM
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
Mr. Sabas Abuabara
Dr. Luis G.C. Rego*
*Current Address:
Physics Department, Universidade Federal do Parana, CP 19044, Curitiba, PR, Brazil,

2. Photo-Excitation and Relaxation Processes
Add energy diagram here
Interfacial electron transfer
Hole relaxation dynamics

3. Aspects of Study Interfacial Electron Transfer Dynamics
Relevant timescales and mechanisms
Total photo-induced current
Dependence of electronic dynamics on the crystal symmetry and dynamics
Hole Relaxation Dynamics
Decoherence timescale.
Effect of nuclear dynamics on quantum coherences, coherent-control and entanglement.
There is some controversy over the appropriate timescales associated with the electron injection mechanism. There are experimental results of short (~100 L.G.C. Rego and V.S. Batista, JACS 125, 7989 (2003)
) and ultrashort(~10 fs) timescales associated with the electron transfer investigated here. Furthermore, there are reports of anisotropy in the transport properties of electrons in the TiO2 bulk and the investigation of whether this effect is manifested in surface reactions is important. How long the initial electron wavepacket remains coherent is also a basic question with ramifications for the robustness of the photocurrent in photovoltaic devices with respect to the temperature and presence of impurities in the crystal structure.
In addition to electron transfer being relatively poorly understood, these studies have important ramifications for the manufacture of photovoltaic cells.
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, (2003)

4. Model System – Unit Cell
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
Catechol molecule adsorbed onto (010) surface of anatase crystal.

5. Ab Initio DFT-Molecular Dynamics Simulations VASP/VAMP simulation package Hartree and Exchange Correlation Interactions: Perdew-Wang functional Ion-Ion interactions: ultrasoft Vanderbilt pseudopotentials

6. Phonon Spectral Density
O-H stretch, 3700 cm-1
(H capping atoms)
C-C,C=C stretch
1000 cm-1,1200 cm-1
C-H stretch
3100 cm-1
TiO2 normal modes
cm-1

7. Electronic Density of States (1.2 nm particles)
LUMO,LUMO+1
HOMO
Band gap =3.7 eV
Conduction Band
Valence Band
ZINDO1 Band gap =3.7 eV
Exp. (2.4 nm) = 3.4 eV
Exp. (Bulk-anatase) = 3.2 eV

8. Mixed Quantum-Classical Simulations Electronic Relaxation
An accurate description of charge delocalization requires simulations to be carried out in sufficiently extended model systems. Simulations in smaller clusters (e.g., 1.2 nm nanostructures) are affected by surface states that speed up the electron injection process, while the implementation of periodic boundary conditions often introduces artificial recurrencies (back-electron transfer events).

9. Model System – Mixed Quantum-Classical Simulations
Three unit cells along one planar directions with periodic boundary conditions in the other.
[-101]
Three unit cells extending the system in
[-101] direction
Due to finite size effects in results from unit cell dynamics calculations, for the purposes of calculations, the actual system consisted of typically 3 unit cells arranged as shown with periodic boundary conditions in the non-extended planar direction.
The PBCs are only used when constructing the Hamiltonian and Overlap Matrices in the extended Huckel formalism.
[010]
System extened in the [010] direction

10. Mixed Quantum-Classical Dynamics Propagation Scheme
, where
and with
the instantaneous MO’s, ..
which are obtained by solving the Extended-Huckel generalized eigenvalue equation :

11. Propagation Scheme cont’d..
where 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 and 1s AO’s of H atoms (i.e., 596 basis functions per unit cell). S is the overlap matrix in the AO’s basis set.
Short-Time Approximate Propagation Scheme
We consider the following AOs:
2s, 2p for O-- (4e-)
2s, 2p for C (4e-)
1s for H (1e-)
3d, 4s, 4p for Ti++++ (e-)

12. Time-Dependent Propagation Scheme cont’d
Assuming a time step so small that the Hamiltonian is time-independent throughout it,

13. Time-Dependent Propagation Scheme cont’d
Setting equal to and multiplying by a MO at the iterated time gives the expression
Which in the dt limit we could be further approximated as
The approximation here amounts to an infinitessimal time increment where the MOs are essentially orthogonal (since they computed by diagonalizing the Hamiltonian at a particular time in the basis of AOs, the eigenvectors at that time are orthogonal).

14. Propagation Scheme cont’d
Therefore, we can calculate the wavefunction and electronic density for all t>0 and we can also define the survival probability for the electron to be found on the initially populated adsorbate molecule

15. Injection from LUMO (frozen lattice, 0 K)
TiO2 system extended in [-101] direction with PBC in [010] direction

17. LUMO Injection (frozen lattice) cont’d

18. LUMO Injection (frozen lattice) cont’d
PMOL(t)
This plot suggests the presence of two distinct timescales for the process of interfacial electron transfer from the catechol LUMO: initial electron injection, followed by anisotropic delocalization within the crystal.

19. Injection from LUMO+1 (frozen lattice, 0 K)

21. LUMO+1 Injection (frozen lattice) cont’d
PMOL(t)

22. LUMO Injection at Finite Temperature (100 K)
PMOL(t)
0 K
100 K

23. Electron Injection from LUMO (100 K)
ln [PMOL(t)]
System extended in [-101] direction represents average over 83 randomly sampled initial conditions.
System extended in [010] direction represents average over 94 randomly sampled initial conditions.

24. Electron Injection at 100 K cont’d [-101] system
ln [PMOL(t)]
The injection time for this system is much faster than for the corresponding static system and there appears to be no distinction of separate timescales.

25. Influence of Phonons on Electron Injection
Representative Nuclear Trajectory cont’d [-101] system; effect of motion on same initial cond’s
The injection time for this system is much faster than for the corresponding static system and there appears to be no distinction of separate timescales.

26. Influence of Phonons on Electron Injection cont’d [-101] system; effect of motion on same initial cond’s
t = 2 fs
The injection time for this system is much faster than for the corresponding static system and there appears to be no distinction of separate timescales.

27. Influence of Phonons on Electron Injection cont’d [-101] system; effect of motion on same initial cond’s
t = 5 fs
The injection time for this system is much faster than for the corresponding static system and there appears to be no distinction of separate timescales.

28. Influence of Phonons on Electron Injection cont’d [-101] system; effect of motion on same initial cond’s
t = 7 fs
The injection time for this system is much faster than for the corresponding static system and there appears to be no distinction of separate timescales.

29. Influence of Phonons on Electron Injection cont’d [-101] system; effect of motion on same initial cond’s
t = 10 fs
The injection time for this system is much faster than for the corresponding static system and there appears to be no distinction of separate timescales.

30. Influence of Phonons on Electron Injection cont’d [-101] system; effect of motion on same initial cond’s
t = 12 fs
The injection time for this system is much faster than for the corresponding static system and there appears to be no distinction of separate timescales.

31. Quantum-Entanglement and Coherent-Control of Hole-Relaxation Dynamics Localized Deep in the Semiconductor Band Gap
t=0 ps
t=15 ps
Super-exchange hole transfer
11/22/2018

32. Coherent Hole-Tunneling Dynamics

33. Investigation of Coherent-Control
t = 200 fs, w12
t= k*t A2 A1 CB
w12
superexchange VB
TiO2 semiconductor
Adsorbate molecules (A1, A2,…)
Agarwal et. al. Phys. Rev. Lett. 86, 4271 (2001)

34. Investigation of Coherent-Control cont’d
2-p pulses (200 fs spacing)
14 fs
60 fs

35. Investigation of Coherent-Control
2-p pulses (200 fs spacing)
2 fs
42 fs

36. Investigation of Quantum Coherences
Reflect entanglement in a change of notation for occupancy
Thus these kets describe the state of all three adsorbates -- at once, i.e., the state of the hole as distributed among all three adsorbates.

37. Investigation of Entanglement cont’d
Example:
In our approximate representation,
represents a physical state of
maximal entanglement
between the center and right adsorbates.
a state defined by only two nonzero expansion coefficients

38. Investigation of Coherences cont’d
Reduced density matrix within the mixed quantum-classical model:
Where the index x indicates a particular initial nuclear configuration

39. Investigation of Coherences cont’d
A measure of decoherence: +

40. Investigation of Coherences cont’d
Compute the subspace density matrix explicitly

41. Investigation of Coherences cont’d
Off-diagonal elements are indicative of decoherence
if nuclear motion randomizes the phases, i.e, becomes a random quantity and the average
The system will no longer be in a coherent superposition of adsorbate states.

42. Investigation of Coherences cont’d
Off-Diagonal Elements of the Reduced Density Matrix

43. Decoherence Dynamics cont’d

44. Investigation of Coherences cont’d
Diagonal Elements of the Reduced Density Matrix

45. Conclusions
We have investigated interfacial electron transfer and hole tunneling relaxation dynamics according to a mixed quantum-classical approach that combines ab-initio DFT molecular dynamics simulations of nuclear motion with coherent quantum dynamics simulations of electronic relaxation.
We have found that an accurate description of charge delocalization requires simulations to be carried out in sufficiently extended model systems. Simulations in smaller clusters (e.g., 1.2 nm nanostructures) are affected by surface states that speed up the electron injection process, while the implementation of periodic boundary conditions often introduces artificial recurrencies (back-electron transfer events).
We have shown that the reaction mechanisms as well as the characteristic times for electron injection in catechol/TiO2-anatase nanostructure are highly sensitive to the symmetry of the electronic state initially populated in the adsorbate molecule.
We have shown that electron injection from catechol LUMO involves a primary step within 5 fs of dynamics that localizes the injected charge on the dxz orbital of the penta-coordinated Ti4+ ion next to the adsorbate (coordination complex ligand mechanism).

46. Conclusions cont’d
We have shown that the 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.
We have found that, 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.
We have shown that the anisotropic nature of carrier relaxation as well as the overall injection process are significantly influence by temperature, since electron-phonon scattering induces ultrafast electron transfer along the mono-layer of adsorbate molecules.

47. Conclusions cont’d
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.
Finally, we have shown that it should be possible to coherently control superexchange hole-tunneling dynamics under cryogenic and vacuum conditions by simply applying a sequence of ultrashort 2p-pulses resonant to an auxiliary transition in the initially populated adsorbate molecule.

48. Acknowledgment Thank you !
NSF Nanoscale Exploratory Research (NER) Award ECS#
NSF Career Award CHE#
ACS PRF#37789-G6
Research Corporation, Innovation Award
Hellman Family Fellowship
Anderson Fellowship
Yale University, Start-Up Package
NERSC Allocation of Supercomputer Time
CNLS Workshop Organizing Committee at LANL
Thank you !