Charge separation and transport in thirdgeneration hybrid polymer-fullerene solar cells.

B.L. Oksengendler, O.B. Ismailova, M.B. Marasulov, N.N. Turaeva, D.U.Matrasulov and J.R. Yusupov

Abstract Exciton dissociation in polymer-fullerene hybrid organic solar cells isstudied within the quantum mechanical and statistical approaches. The differentmechanisms for splitting of exciton into electron and hole is discussed.

1 Introduction

Third generation solar cells based on organic photovoltaic materials are being con-sidered as a serious alternative for silicon based ones The bulk heterojunction Usu-ally such solar cells are fabricated using polymer structures as donors and fullereneembedded into polymers as acceptor. In some cases quantum dots can be embeddedinto the polymer matrix. The maximum conversion efficiency of such solar cellsare expected to reach 80% [6] due to so-called multiple exciton generation effect[9]. The mechanism such effect is still subject for discussions and extensive studies.Topics of the past studies of third generation solar cells both polymer-fullerene andpolymer-quantum dot based ones can be classified as follows:

B.L.OksengendlerInstitute of Polymer Chemistry and Physics, Uzbekistan

O.B. IsmailovaInstitute Ion-Plasma and Laser Technology of Uzbekistan Academy of Sciences, Uzbekistan

M. B. MarasulovInstitute of Polymer Chemistry and Physics, Uzbekistan

N.N. TuraevaBiological Department, Webster University, USA

D.U. MatrasulovTurin Polytechnic University in Tashkent, Uzbekistan

J.R. YusupovTurin Polytechnic University in Tashkent, Uzbekistan

1

2 B.L.Oksengendler et al.

1. Mechanism of the multiple exciton generation by quantum dots induced by singlephoton absorption;

2. Transmission of generated excitons through the quantum-dot-polymer or polymer-fullerene interface and possible splitting of exciton into electron and hole duringthis separation;

3. The exciton transport and charge separation along the polymer chain;4. Transport of electrons and holes through in the polymer-quantum dot or polymer-

fullerene networks;5. The charge carrier collection at the metal-polymer interface;6. Degradation of solar cell.

Among the above issues (2), (3), and (6) are still remaining as less studied, de-spite the extensive studies during last few years [6, 2, 4, 9, 12, 3, 8].

In this paper we develop microscopic mechanism for exciton splitting and chargetransport in organic solar cells on the basis of polymers and nanoparticles (e.g. quan-tum dot, fullerene).

2 Charge separation via dopant charge exchange

Assuming that electron is moving along linear polymer chain and the mass of thehole is large enough, splitting of the exciton can be considered as similar to chargeexchange between the hydrogen atom and a potential well induced by effective pos-itive charge (Fig. 1).

Fig. 1 Schematic representation of exciton dissociation via resonance rechargeable.

Charge separation in third generation solar cells 3

In [10] the process of exciton dissociation on fullerene via the reaction

ex+F0→ h+F−, (1)

was considered as a resonance charge exchange studied in [13]. However, in generalcase potential wells induced by electron and hole on fullerene trap have differentdepths. Therefore one needs to strict treatment of charge separation as a chargeexchange process. This can be done within the Landau-Zener theory [5].

If exciton moves along linear polymer chain and approaches to fullerene molecule,the molecular terms corresponding to states ”electron-on hole+fullerene” and ”electron-hole on fullerene” (Fig. 1) can be written as{

U1(R) =U (1)0 +∆Eex

U2(R) =U (2)0 − e2/εR

(2)

where U (1)0 and U (2)

0 are the terms of electron on polymer (highest occupied orbital,HOMO) and trapped into fullerene correspondingly; ∆Eexis the exciton perturbationenergy, ε is dielectric constant of polymer matrix.

As it can be seen from Fig. 2, the terms U1(R) and U2(R) are crossing at thepoint:

R∗ =e2/ε

U (2)0 −U (1)

0 −∆Eex

(3)

In its motion along the polymer chain with the velocity υ , exciton passes throughthe point R∗ (Fig. 2) that leads to radiationless transition from term 1 to 2 (Landau-Zener transition) and subsequently the reaction given by Eq.(1) occurs. Furthermore,the ”free hole” continues to move along the chain and reaches the point R∗. If theelectron hole system remains at the term 2 this will correspond to the splitting ofexciton into electron (which will remain at the fullerene) and hole that will movealong the chain. Then the probability for such decay can be written as [5]:

Fig. 2 Schematic representation of the crossing terms V1(R) (exciton + F0) and V2(R) (hole + F−)

4 B.L.Oksengendler et al.

P(υ) = ω1→2′(R∗)[1−ω2′→1(R

∗)]

(4)

ω1→2′(R∗) = ω2′→1(R

∗) = 2

[1− e

− 2π|V |2hυ|F1−F2|

]e− 2π|V |2

hυ|F1−F2| , (5)

where F1(R∗) and F2(R∗)

are values for forces of terms 1 and 2 at the crossing point

R∗ and R∗,∣∣V ∣∣2 is the matrix element describing transition from term 1 to term 2 in

second order of perturbation theory; υ = υ√

1−ρ2/R∗2 is the velocity componentdirected along fullerene (Fig. 2). Using Eqs. (4) and (5), for the of exciton splittingcross -section we have

σdecex (υ) = 2π

∫ R∗

0P [υ(ρ)]ρdρ (6)

if the perturbation is week enough, i.e.,∣∣V ∣∣2 << hυ |F1−F2| , by taking into account

Eqs. (4) and (5) from Eq. (6) we obtain

σdecex ≈

8π2∣∣V ∣∣2 (e2/ε

)3

h[U (2)

0 −U (1)0 −∆Eex

]41υ

(7)

As it can be seen from this result, exciton splitting cross section depends on (υ)and (∆Eex). This fact imposes certain restrictions for using of polymer matriceswith high mobility of excitons. Having known the cross section, σdec

ex for excitonsplitting one can easily estimate the relaxation time for emitted excitons [13]:

τdecex = 1/σ

decex NF ·υ , (8)

here NF is the concentration of fullerene molecules in the polymer matrix (net-work).

3 Statistical model for exciton dissociation in regular polymers

One of the possible mechanisms for splitting of exciton into electron and hole is intransition via the polymer network vertex. If one draws a sphere of ρ˜h/

√2m0∆Eex

around the polymer vertex where lying at the crossing N bonds of the polymer chain,it follows from the uncertainty principle that exciton can be in dissociated stateinside such sphere. To some extent such a state of analog of well-known compoundstates from nuclear physics [1]. The life time of such state can be estimated as [5,1] τkc ≈ h/∆Eex. After this time exciton can pass through the vertex as a wholesystem or splits into electron and hole which will move along different bonds of apolymer chain (Fig. 3). The probability for splitting of exciton at the vertex withinthe statistical physics based approach can be written as [7]

Charge separation in third generation solar cells 5

Fstat = Zsep/Zsum = (2C2Ne−∆Eex/kT )/[N +2C2

Ne−∆Eex/kT ], (9)

where are Zsep and Zsum are the partition functions of splitted and whole systems,respectively.

For ∆Eex/kT << 1 Eq.(9) can be written as

Fstat =N−1

N. (10)

It is clear from this expression that in large N limit the probability approaches 1. ForN = 3 we have Fstat = 2/3.

For ∆Eex/kT >> 1 from Eq. (9) we get

Fstat = 2(N−1)exp(−∆Eex/kT ) (11)

Having known the probability for exciton splitting, Fstat from Eqs. (9), (10) and (11)one can get estimate for the exciton dissociation time as

τcsex = 1/σ

csex ·N0υ ·Fst = 1/N0υπ

(h√

2m∆Eex

)2

·Fstat , (12)

where N0 is the concentration of excitons in polymer matrix.For N0 = 1021sm−3, υ = 105sm/s, ∆Eex ≈ 5e, V N = 3 we have τcs

ex = 1.5×10−11s.

Fig. 3 The exciton separationat the polymer chain vertexconnecting three bonds)

4 Quantum mechanical model: Perturbation theory

Constructing of quantum-mechanical model for exciton dissociation in its transmis-sion through the polymer network vertex is of importance for understanding mi-

6 B.L.Oksengendler et al.

croscopic mechanisms of the process. In this work we develop perturbation theorybased approach for such study . The main question we are interested to explore is:What kind of explicit form has the perturbation potential acting to the exciton at thepolymer chain vertex?

We assume that the potential acting to exciton at the vertex of the polymer chainhas delta-function form:

U =U0a3δ (→r ). (13)

where U0, a3 and δ are measured in eV , A3 and cm−3, respectively. For the vertexhaving N - bonds this potential can be written as V = N×U .

Furthermore, we consider exciton moving along a bond of the polymer chain andapproaching a vertex of the chain. Fixing the origin of the coordinate system at thecenter of mass of exciton one can consider interaction of the exciton with exter-nal potential and a scattering of a vertex on exciton. Introducing two characteristictimes, reorganization time of the exciton, (τ in) and the time during which interac-tion with a vertex occurs, (τext ), we can consider two limiting situations for whichwe can use different types of perturbation theory. Namely, [5] for τin >> τext , onecan use sudden perturbation approximation, while for the case τin << τext , one canapply adiabatic perturbation theory.

To find in under which conditions each regime is possible we need to estimatethe time during which exciton reaches the vertex of the polymer chain, i.e. the areawhere external perturbation acts. If the motion of exciton from bond to bond isjump-like motion, then each jump is characterized by diffusion coefficient whichcan be written as [12],

D∼=16

a2/τ j, (14)

where 1/τ j is the jump frequency. On the other hand, within the Bohr atom frame-work, the internal reorganization time can be estimated as

τin =2π√

r3µ

e, (15)

where r is the effective exciton radius, µ is the reduced mass of exciton 1µ= 1

me+ 1

mn,

τin = 1.8×10−16ε

(m0µ

)c, dielectric permittivity ε of the polymer, m0 is the electron

mass, τext = 1.8×10−16 1D c. Thus the boundary between two regimes can be defined

as

ε

(m0

µ

)=

1D

(16)

(16): The ”phase diagram” describing these regimes can be represented as in theFig. 4.

Consider the case when the following condition is obeyed:

1D

< εm0

µ. (17)

Charge separation in third generation solar cells 7

Potential acting to exciton at the vertex connecting N-bonds of the polymer chaincan be written as

Uvertex = N×U0a3δ

→(r)

Then the probability for transition of exciton from the ground state, ϕground =√π

a3 e−r/na to an excited state, ϕex =√

π

a3n3 e−r/na can be written as

Wgr→ex =

∣∣⟨ϕex |Uvertex|ϕground⟩∣∣2

(hω)2 =|M|2

|hω|2, (18)

where hω ≈ E(n)ex −E(gr)

ex .We assume that the exciton shaking processes consist of two stages: Excitation

of the electron in exciton into the higher orbit and its transition into conductanceband induced by thermal motion. Subsequently, the hole appears in lower band.Such transition is possible under the condition

∣∣∣E(n)ex

∣∣∣ ≈ kT , and we have hω =

−∆Eexn2 +∆Eex = ∆Eex

(1− 1

n2

). Using the wave functions of ground and excited

states, the matrix element in the expression for the transition probability can bewritten as

M =⟨ϕex |Uvertex|ϕgr

⟩=

π

a3 U0N1

n3/2 (19)

Then for probability we have

Wgr→ex = π2(

NU0

∆Eex

)2( kT∆Eex

)3/2(

11− kT

∆Eex

)2

, (20)

Fig. 4 ”Phase diagram” based on equation (16)

8 B.L.Oksengendler et al.

Fig. 5 Schematic representation of exciton dissociation and association via scattering at the vertex.

The total probability for dissociation of exciton into electron and hole being dis-tributed over the bonds of the polymer chain can be written as

W =(Wgr→ex

) N−1N

(21)

In case of the adiabatic perturbation occurring under the condition

1D

> εm0

µ, (22)

one can use Landau-Zener theory [5] dissociation theory. For this purpose we needto plot the energy terms of the systems ”exciton-vertex” and ”unbounded electron-hole-vertex”(Fig. 5). To find the explicit functional forms of these terms we assumethat they can be considered as the sum of the ground state energy of exciton andVan der Waals interaction between the exciton and chain vertex. The latter will bemodeled as the dielectric ball. Assuming that the Van der Waals interaction is weakenough we can plot the terms presented in (Fig. 5).

The potential of interaction between the vertex and the free carriers can be con-sidered as a result of the polarization of electron and hole in dielectric medium(Fig. 6).

Then the effective electrostatic field caused by electron-hole pair and acting on avertex can be written as [14]:

|E++E−|=1ε

2qcosα

R2 + l2/4=

12ε0

2ql

(R2 + l2/4)3/2 . (23)

It is clear that in case of separation of electron and hole over the different bondsof the chain, one can write l

2 = Rctgα . For this case the energy term can be writtenas

Charge separation in third generation solar cells 9

E(e+h)ex (R) = −1

2αpol |E++E−|2−

e2

R×2ε× ctgα

= −12

αpol

(ε)24e2 cos2 α

R4 (1+ ctg2α)2 −e2

R×2ε× ctgα, (24)

where apol is the vertex polarization angle α depends on the number of bondsconnected at the vertex:

α = 90◦(

1− 2N

)(25)

The term illustrated in Fig. 5 crosses with that of delta -like potential at the R∗:

R∗ ≈(

ecosα

1+ ctg2α

)1/2(

2αpol

∆Eex(∞)

)1/4. (26)

The probability for inirradiative Landau-Zener transition describing of excitondissociation at the point R∗ can be written as [5]:

1−W2 = 1− 4πV 2

hν |F2|, (27)

where V 2 is the square of the adiabatic coupling operator matrix element, ν is ve-locity of exciton

F2 =

(∆Eex(∞)

2

)5/4 8

α1/4pol (ε)

2

(1+ ctg2α

ecos2 α

)1/2− (28)

second term force at the point R∗. Furthermore, for the case when exciton amovesalong the chain the crossing point of terms can be found from the following equa-tion:

Fig. 6 Polarization of thevertex B, that is consideredmodeled as a dielectric ball.

10 B.L.Oksengendler et al.

Uoa3δ (R)−∆Eex (∞) =−12

αpol12

2ql0(R2 +

l204

)3/2 , (29)

where l0 is the dipole moment of the electron-hole pair in two electron bands.The total exciton dissociation probability can be written as a product of two prob-

abilities asW =W1× (1−W2), (30)

where W1 =4π2|V |2

hυ |F(R∗∗)| is the probability for transfer from one term to another one atthe point R∗∗ and (1−W2) is the probability for staying of free electrons and holesat the point R∗.

5 Quantum mechanical model: Quantum graphs based approach

An alternative to the above approaches for description of exciton splitting in poly-mer network is so-called quantum graph based description. In physics quantum

Fig. 7 Gaussian wavepacket evolution in quantum star graph. The bonds of the graph have differentlengths.

Charge separation in third generation solar cells 11

Fig. 8 Gaussian wavepacket evolution in quantum star graph. The same graph in Fig. 8 with thedifferent initial velocity of the packet and packet width.

graphs were introduced as a ”toy” model for studies of quantum chaos by Kot-tos and Smilansky [15]. However, the idea for studying of a system confined to agraph dates back to Pauling [18] who suggested to use such systems for model-ing free electron motion in organic molecules. During last two decades quantumgraphs found numerous applications in modeling different discrete structures andnetworks in nanoscale and mesoscopic physics(e.g., see reviews [15]-[17] and ref-erences therein). However, mathematical properties of quantum graphs were exten-sively studied in eighties of the last century. Later quantum graphs became subjectfor extensive research in different topics of mesoscopic and nanoscale physics andquantum chaos theory (see, e.g. review papers [15]–[17] and references therein).

Graphs are the systems consisting of bonds which are connected at the vertices.The bonds are connected according to a rule which is called topology of a graph.Topology of a graph is given in terms of adjacency matrix [15, 16]:

Ci j =C ji =

{1, if i and j are connected0, otherwise i, j = 1,2, ...,V.

Quantum dynamics of a particle on a graph is described by one-dimensionalSchrodinger equation [15, 16] (in the units h = 2m = 1):

12 B.L.Oksengendler et al.

i∂Ψb(x, t)

∂ t=−∂ 2Ψb(x, t)

∂x2 +V (x, t)Ψb(x, t) (31)

where b denotes a bond connecting ith and jthe vertices, and for each bond b, thecomponent Ψb of the total wavefunction Ψb is a solution of the Eq.(31).

In this work we consider simplest topology, so-called star graph, i.e. three ormore bonds connected at the single vertex. In this case the boundary conditions canbe written as

φ1|y=0 = φ2|y=0 = ...= φN |y=0,φ1|y=l1 = φ2|y=l2 = ...= φN |y=lN = 0,

N∑j=1

∂

∂y φ j|y=0 = 0.(32)

We note that to some extent, quantum graphs can be convenient and effectivetool to describe charge separation and transport in polymer networks which are theunderlying structure of organic solar cells. In particular, exciton can be describedas wave packet and splitting of exciton into electron and hole can be easily treatedwithin such approach. The Schrodinger equation can be solved with the initial con-dition given in the form of Gaussian wave packet:

ψ1 (x1, t = 0) =1√√2πσ

e−(x−µ)2

2σ2 +ip0x. (33)

Fig. 8 presents wave packet dynamics (which effectively can describe exciton dy-namics) on quantum star graph with three bonds. The profile of the packet is plottedfor different time moments. Sensitivity of the dynamics with respect to the changesof the initial velocity of the packet and packet width can be seen from the Fig. ??.Using more realistic initial wave function, i.e. wave function of exciton allows tomake the results more realistic. Also, including into the Schrodinger equation thepotential for interaction of exciton with the vertex should improve the quality of theobtained results. Thus quantum graphs can be rather effective tool for descriptionof the exciton dynamics and charge separation in polymer networks. Treating theprocess for other (than star graph) more complicated topologies is of importance forpractical application of the method to poly-conjugated polymers.

6 Conclusion

A microscopic mechanism of the exciton dissociation in the polymer network istreated within the quantum mechanical approaches. Considering exciton as an ana-log of hydrogen atom allows one to consider exciton separation as ionization pro-cess occurring in the ”collision” of exciton with the vertex of the polymer chain. Inthis case, depending on the ”collision energy” sudden- and adiabatic perturbationtheories can be used to describe splitting of exciton at the chain vertex. Alterna-tive approach, which is based on modeling of polymer network by quantum graphs

Charge separation in third generation solar cells 13

provides possibility to study the dependence of the dissociation process on the net-work/chain topology.

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