A Method to Rapidly Predict the Injection Rate in Dye Sensitized Solar Cells.

Daniel R. Jones and Alessandro TroisiDepartment of Chemistry and Centre for Scientific Computing,

University of Warwick, Coventry, UK

www.warwick.ac.uk/go/troisigroup

1. AbstractWe present a technique which can be used to predict the rate of electron transfer between a chromophore molecule and the TiO2 surface in a dye sensitized solar cell (DSSC). Using the Green’s Function operator and the diabatisation of the system into molecular and semi-conductor states we are able to predict, without additional parameters, the rate of ultrafast injection from the first excited state of the chromophore into the TiO2 conduction band. By partitioning the system into three subsystems, the TiO2 surface, the chromophore and their interface, we are able to significantly reduce the expense of the DFT computation allowing quick access to the rate of electron transfer with modest computational cost.

2. IntroductionSince its inception in 1991 the DSSC has attracted significant interest as a potential source of affordable renewable energy. However, despite the promising efficiency of the early devices there has been little improvement since early prototypes, the maximum efficiency devices have about 12% incident photon to current efficiency.

A computational method which allows a rapid prediction of the rate of important electron transfer processes in the DSSC could enable new progress in the development of more efficient dyes and devices.

4. Computing the rate of ETThe system of chromophore and TiO2 surface is modelled using an effective Hamiltonian constructed starting from,

(1)

Where are states on the chromophore and states in are states on the surface. This can be represented in the S subspace by an energy dependent effective Hamiltonian,

(2)

The self energy, Σss’, can be separated into real and imaginary components, and the imaginary component of the self energy is equal to the rate of decay of the state. We denote the imaginary component of self energy, Γ. Using the Kohn-Sham Fock matrices computed for the systems in section 3, Γ is computed on an atomic basis set using,

(3)

Where Vmk is computed using the interface between the surface and molecule from subsection (c) and ρkk’ is the energy dependent density matrix computed based on the surface electronic structure using,

(4)

Where Ckl is the matrix of eigenvectors that diagonalizes the Fock matrix of the surface and δ is the Dirac delta function.

Load Voltage

Conductive Glass Electrode

3 I−

Dye CoatedNanocrystalline TiO2

CounterElectrode

I3−

3. Partitioning the SystemTo compute the rate of electron injection from the chromophore into the TiO2 surface the system (a) was partitioned into three sub-systems. The chromophore molecule (b), the interface between the chromophore molecule and the surface (c) and TiO2 surface (d).

The electronic structure of the complete system was approximated using a model Hamiltonian constructed using the Kohn-Sham Fock matrix of the 3 sub-systems computed using the B3LYP density functional and a 6-31G* basis set in Gaussian 03. For these computations the chromophore molecule was protonated and the TiO2 surface was modelled using a cluster ((TiO2)35

for anatase (101) and (Ti32O62)4+ for rutile (110) embedded in a large volume of point charges 2+ in place of Ti atoms and 1− for O atoms. The DFT computations for subsystems (c) and (d) can be reused for many different chromophores making the method computationally inexpensive.

1 1 1 1

.n M n M

s l sl lss l n s l n

H s E s l E l V s l V l s

s l

' 0 '( ) ( ).effss ssH E H E

*' '

'

2( ) ( ).mn mk k n kk

kk

E V V E

*' '( ) ( ).kk kl k l l

l

E C C E E

5. Preliminary Results and ContinuationThis matrix can be rotated onto the LUMO of the chromophore at an appropriate injection energy from TD-DFT to compute a rate of injection.

(5)

We plot the rate of injection from the model chromophore perylene at various values of injection energy

Using this method we compute injection rates of between 1 and 20 fs for some recently reported organic chromophores. These results compare favourably with experimental transient absorption spectroscopy measurements.

We plan to extend this work to include chromophores coupled by alternate anchoring groups and also to investigate the rate of the recombination reaction.

,

( ) ( )r mn rm rnm n

E E c c

E in this range

Perylene LUMO

(a)

(b)

(c)

(d)