Conclusiones finales del capítulo

recombination faster than in n-type devices?

3.1 Introduction

The development of p-type dye-sensitized solar cells (p-type DSSCs) was motivated by the recognition of the high theoretical efficiency that could be achieved by tandem cells with two photo-active electrodes,50 as introduced in Chapter 1. In such types of device the generation of photo-current is limited by the weaker electrode, which is the photo-cathode that operates in accordance with the principles of p-type DSSCs (Fig. 1.2). Understanding the origin of the weak photocurrent generation in p-type DSSCs was therefore deemed important for improving the performance of tandem DSSCs.

The primary charge generation process in p-type DSSCs is promoted by a photo-excited dye that injects a hole into the semiconductor. Hole injection is the transfer of a hole from a localized orbital of the dye, typically the HOMO, to a one-electron state of the VB of the semiconductor. The analogous process in n-type DSSCs is the electron injection from the LUMO of the dye to the CB of the semiconductor. Hole recombination, on the other hand, is the transfer of a hole from a one-electron state of the valence band of the semiconductor to an orbital of the dye, typically the LUMO, or it could be more intuitive to visualize as an electron transfer from the LUMO of the dye to an empty orbital in the VB. This process is an analogue to charge recombination in n-type DSSCs, where electrons back-transfer from the CB of the semiconductor to the HOMO of the dye. Charge recombination processes contribute to the decrease of the solar cell

efficiency and should be minimized.

While hole injection in p-type DSSCs appeared to be as efficient as electron injection in n-type DSSCs, hole recombination was considered as one of the main causes for the low photo-current observed in p-type DSSCs.208 A number of experimental investigations on the hole transfer kinetics in p-type DSSCs demonstrated that the hole recombination was much faster than desirable. TAS and kinetic analysis of Coumarin 343 (C343) sensitized NiO showed that, while charge injection had an ultrafast component and was similar to common n-type DSSCs,209,210 charge recombination occurred at the timescale of tens of pico-seconds,56,211 and it was faster than the recombination times observed in C343 sensitized TiO2. Electrochemical impedance spectroscopic studies also

addressed the charge recombination problem in NiO p-type DSSCs.60,212 Attempts to suppress recombination included adjusting molecular dipole alignment on NiO surface,212 increasing the tunneling distance by extending oligo-thiophene linker in donor-π-acceptor dyes,51 and substituting conventional I3-/I- redox shuttle with alternatives such as the Co(II)/(III) pair.52 While some

success on retarding recombination had been achieved, such as slowing down the process from ns to μs,51,213 the extent of improvement was insufficient for producing p-type DSSCs with PCE comparable to that of n-type DSSCs. The origin of the fast recombination had in fact remained unclear, and intuitive suggestions on retarding recombination might negatively influence other processes in the device, such as hole injection,213 hence resulting in small improvement of PCE even with slower recombination. In this chapter we attempt to explain why recombination is faster in p-type DSSCs than in n-type DSSCs. Based on the injection and recombination theories introduced in section 2.2, we evaluate the hole injection and recombination rates at the NiO-C343 interface, a

typical semiconductor-dye interface in p-type DSSCs, and compare results with analogous interface in n-type DSSCs.

3.2 Theoretical background and computational methods

3.2.1

Theory of hole injection

The method for computing the rate of hole injection in this chapter is essentially identical to that for computing electron injection in n-type DSSCs (see section 2.2.1) except for the orbitals involved.65 The rate (Γss(E)) of a hole transfers from

a one-electron state s , such as the HOMO of the dye, to a manifold of states

{ l }, such as an orbital in the VB of the semiconductor, can be expressed as Eq.

2.9. In this case, in Eq. 2.9, E is the energy of the HOMO of the dye, El is an

eigenvalue of the manifold of eigenstates in the semiconductor, such as the valence band (VB) maximum; Vsl is the coupling between state s and state

l and δ(E − El) is the Dirac delta function, which in this work is approximated

by a normalized Gaussian function with 0.1 eV broadening. The states s and l are conveniently represented as linear combinations of basis functions {χm}

and { k} localized on the dye and the semiconductor respectively:

ms m m s c (Eq. 3.1) kl k k l a (Eq. 3.2) where {cms} and {akl} are the orbital coefficients derived from the calculations of

isolated dye and semiconductor respectively. The coupling Vsl can be expressed

in terms of the coupling Vmk between localized orbitals χm and k:

sl ms kl m k

m k

V c a V (Eq. 3.3) and the rate can be expressed as:

* , ( ) ( ) ss mn ms ns m n E E c c (Eq. 3.4) where * * ' ' , ' 2 ( ) ( ) mn mk nk kl k l l l k k E V V a a E E (Eq. 3.5)

The equation above can be re-written by defining the local density of states ρkk’(E)

as: * '( ) ' ( ) kk kl k l l l E a a E E (Eq. 3.6) which gives Eq 2.11. For non-orthogonal basis sets, Γmn(E) should be modified to

become: * ' ' ' , ' 2 ( ) ( )( ) ( ) mn mk mk nk nk kk k k E ES V ES V E (Eq. 3.7)

In practice, to compute the charge injection rate one needs (i) the energies, El, of

the considered NiO Kohn-Sham eigenstates, l , relative to the HOMO energy E;

(ii) the coefficients of the basis functions of NiO, alk, and of C343, cms; (iii) the

coupling term Vmk; and (iv) the overlap matrix element Smk.

3.2.2

Theory of hole recombination

The approach to calculate hole recombination in p-type DSSCs is similar to that for calculating charge recombination in n-type DSSCs.87 The hole recombination rate (kh,rec) is given by:

, ii( )(1 ( F)) ( , p, )p

h rec

k E f E E F E G dE (Eq. 3.8) The first term of the integrand, Γii(E), is given by Eq. 2.13, where the coupling Vil is between one-electron states { l } in the semiconductor and a one-electron

state i in the dye, which corresponds to the LUMO of the dye. Γii(E) can

* , ( ) ( ) ii mn mi ni m n E E c c (Eq. 3.9) In Eq. 3.8, f(E – EF) is the Fermi-Dirac distribution and EF is the Fermi level, or

the quasi-Fermi level in non-equilibrium situations. The term 1 – f(E – EF) is

therefore the probability of a hole occupying a state at a given energy. F(E, ΔGp,

λp) is the thermally averaged Franck-Condon integral between initial and final

vibrational states: 2 ( ) 1 ( , , ) exp[ ] 4 4 p p p p p B p B E G F E G k T k T (Eq. 3.10)

which is similar to Eq. 2.15 and differs most notably with the use of different ΔG

and λ in these expressions, as discussed in section 2.2.3. In practice, to compute the charge recombination rate given by Eq. 3.8 one needs (i) the function Γii(E),

which is calculated with the same procedure used in hole injection, except that the MO coefficients in Eq. 3.9 are those of the LUMO instead of the HOMO of the dye; (ii) the quasi-Fermi level, EF; (iii) λp for Eq. 2.18; and (iv) ΔGp for Eq.

2.18.

3.2.3

Electronic structure calculation of the semiconductor and