научная статья по теме QUANTUM AUTO-IONIZATION OF MOLECULAR EXCITONS AND PHOTOVOLTAIC CONVERSION Физика

Текст научной статьи на тему «QUANTUM AUTO-IONIZATION OF MOLECULAR EXCITONS AND PHOTOVOLTAIC CONVERSION»

Pis'ma v ZhETF, vol. 101, iss. 1, pp. 19-23 © 2015 January 10

Quantum auto-ionization of molecular excitons and photovoltaic

conversion

V. A. Benderskii+, E. I. Kats* x^ +Institute of Problems of Chemical Physics of the RAS, 142432 Chernogolovka, Russia * Landau Institute for Theoretical Physics of the RAS, 142432 Chernogolovka, Russia x Moscow Institute of Physics and Technology, 141700 Dolgoprudny, Russia Submitted 14 November 2014

This paper explores a novel way ol charge separation (auto-ionization) ol molecular excitons, by quantum tunneling through nano-size p—n junction. This mechanism can dominate the standard one (i.e., when Frenkel exciton is ionized at donor or acceptor impurity sites) for very short, nano-size, p—n junction, where the junction electric field can be strong for relatively small (on the order of 1 V) voltage drop. Within a simple one-dimensional model for the depletion region of the p—n junction (donor and acceptor reservoirs connected by a short molecular wire) we compute the quantum yield Yb for the tunneling exciton auto-ionization in the "bulk" of the depletion region. For modern organic photo-sensitive materials with p—n junction size on the order of 10-20 nm, Yb could be close to 1. Such a high efficiency of the charge separation (one of the main factor entering figure of merit, indicating how good are photovoltaic conversion cells) makes this new mechanism potentially very perspective for the applications.

DOI: 10.7868/S0370274X1501004X

Background. Organic semiconductors (like ph-thalocyanines, doped-fullerenes, and some other photosensitive compounds) already long time attract attention of scientists (see, e.g., [1-4]) as potentially perspective materials for solar cell applications. Usually for conventional micron size p—n junction, in the lowest energy state, the electron and the hole (forming the neutral Frenkel exciton) are localized on the same molecular site. Quantum dynamics of such excitons can not lead to charge transfer. Charge separation may occur only when diffusing Frenkel exciton meets the impurity site. Main difficulty for a practical application of such mechanism for photovoltaic conversion is that light adsorption in organic semiconductors is related to intramolecular state transitions (to contrast with inter-band transitions in conventional inorganic solid state materials). Charge transfer by excitons, created by these intramolecular transitions, is related to a very low efficiency process of charge separation. Much research performed in order to enhance the efficiency of charge separation failed because there are conceptual restrictions for such processes in conventional (macroscopic size) heterojunc-tions. The matter is that ionization of the excitons requires some additional energy which can be taken from thermal phonons. The probability of the corresponding

-^e-mail: efim.i.kats@gmail.com

processes is very small in organic semiconductors. However, and this is the main point of our work, there is one more way for free charge carrier formation. Namely, the exciton auto-ionization through the intermediate (appearing at higher energy) charge transfer exciton states (where the e—h pairs are smeared over many sites). Unfortunately, the probability of requiring quantum tunneling processes is very low for a standard macroscopic size p—n junction.

From the experimental point of view the situation turns out less desperate. Recent progress in nanotech-nology [5-7] allows to design nano-size heterojunctions randomly or regularly spaced in photo-sensitive organic materials. However despite the undoubted relevance of a such approach, it suffers from some major drawbacks that limit the practical perspective of the applications. Theoretical estimations [5-7] show that even for the p—n junction size, 20-30 nm, comparable with the localized exciton characteristic diffusion length (Ze ~ \JDere, which is on the order of 10 nm for a typical life time re ~ 10~10s and diffusion coefficient De ~ 10~2cm2/s), the best expected efficiency for all known photo-sensitive organic materials is still smaller than that for the standard solid-state inorganic systems.

In our paper we would like to take a fresh look at the theoretical estimations made for the nano-size p—n junctions in organic photo-sensitive materials. We are

IlHCbMa b >K3TO tom 101 Bbin. 1-2 2015

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y. A. Benderskii, E. I. Kats

discussing a novel way (overlooked in previous considerations) of charge separation (ionization) of molecular excitons, by quantum tunneling through the depletion region of the p—n junction. For the nano-size system this tunneling auto-ionization can lead to much more efficient charge separation than that realized outside the depletion region (at its boundaries within our model, i.e., by "surface" charge separation). Indeed intrinsic internal voltage through the p—n junction is determined by characteristic impurity (donor and acceptor) levels spacing. It is independent of the junction length L. Therefore bulk (by the tunneling ionization) charge separation processes exponentially small (over the parameter le/L) for the conventional macroscopic size hetero-j unctions, can at L ~ le become comparable with the surface (i.e., at the junction boundaries) exciton ionization processes. These very short heterojunctions in photo-sensitive organic materials could be still promising candidates for the efficient charge separation processes needed for solar cells. Writing our paper we hope to motivate new experimental and advanced theoretical studies of the nano-size p—n junctions in organic photosensitive molecules.

Of course no miracles and not everything is so cloudless. To get efficient bulk charge separation is not sufficient to produce very short p—n junction. The length L of the junction has to be larger than electron and hole Debye screening lengths rThe condition to be satisfied reads as

le>L>{rg+r$)}, (1)

where in self-evident notations

/ \ 1/2 (e) ( W) _ / K0kBT \

with njf'^ being the concentration of the main charge carriers (electrons or holes) in the heavily doped regions, and ko — 3—4 is dielectric permeability. For typical in photo-sensitive organic materials [5-7] parameters (impurity ionization energy 0.2 eV, impurity concentration 1019 cm-3), charge carrier concentration njf'^ could be as large as 1017 cm~3 and it gives ~ 5 nm.

Organic photo-sensitive materials we are talking about are formed by flat-shaped molecules packed into layers with relatively weak interlayer coupling. Therefore, it seems naturally to start theoretical investigation with one dimensional model for the p—n junction. While the effects related to realistic three dimensional molecular packing and impurity configurations may lead to certain changes, it is not expected to radically alter

the picture we are studying here. Although pure mathematically our one dimensional case is simpler, but all essential ingredients and difficulties (e.g., competition between exciton tunneling and diffusion) are already there.

Charge separation in one dimensional model for the nano-size p—n junction. With all said above in mind we model the p—n junction by one dimensional finite length L chain of photo-sensitive molecules placed in the sites 1, 2, ..., N of the chain (where N = L/Iq with lo being the lattice period; in what follows, unless opposite will be said we use lo as a unit of the distance). The chain is supposed to be under a constant electric field which models the p—n junction intrinsic internal field. It is convenient to write down the model Hamiltonian in terms of the localized at the molecular sites wave functions. The exciton (electron-hole pair) wave function can be characterized by the sites n+, where the hole and the electron respectively are localized. One can easily see that the quantum dynamics of the exciton center of mass is separated from the relative electron-hole motion we are only interested in. Then, diagonal matrix elements of the Hamiltonian depend only onn = | n+ — n~\ and read as

V(n) = (n+ ,n~\H\n~ ,n+) =

i-Vi(l/n)-V2(n/N), n> 1,

\ -V0, n = 0.

Here n = 0 corresponds to the localized state of the molecular (Frenkel) exciton (the bound state energy is —Vo), and n = 1, 2, ..., N elements determine characteristic distance of the charge separation in the exciton (—V\jn term simulates the electron-hole Coulomb attraction, and —V^n/N term models the intrinsic p—n junction electric field), Vo is the exciton complete ionization energy, and naturally Vo >V\.

We take into account only the nearest neighbors hopping and then, the non-diagonal matrix elements are written in the following form

{n+,n-\H\n~ ± l,n+ ± 1) = en (4)

and

(n+, n~\H\n~, n+ ± 1) = e_|_,

(5)

(n+,n \H\n ±1 ,n+)=e_.

The matrix elements eo (and en with n = 1, 2, ... for electron-hole separation distance n) for the exciton (i.e., e and h in block) transfer are determined by the probabilities of the transitions between HOMO (highest

occupied molecular orbitals) and LUMO (lowest unoccupied molecular orbitals) states [7]. For the localized Frenkel exciton en (inverse hopping time) in organic photo-sensitive molecules is typically en — Iq/De — (0.01—0.03) eV. On the contrary, the elements e± (the widths of the valence and conduction bands respectively) are determined by the nearest sites hopping probabilities either for electron or for hole moving separately. In the natural situation e± > en, and as we mentioned already the uncharged exciton center of mass motion is separated from the relative (in opposite directions) movements of the electron and of the hole.

Transforming the site localized basis into propagating wave basis (with the wave vector k and given e—h separation distance n)

1

N V2

exp[ik(n+— n )]|í?+,í?. }, (6)

the matrix elements (3), (4) are

{iik(n)\H\rk(n)) = V(n) + ]T enexp[^(r>+ + «")], (7)

and

{Mn)\H\ii*k(n ± 1)) = e+ exp(±A-) + e_ exp(TA') =

= efcexp (iôk),

(8)

where following [8] and [9

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