Concept of the Time-Dependent Diffusion Coefficient of Polarons in

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Concept of the Time-Dependent Diffusion Coefficient of Polarons in Organic Semiconductors and Its Determination from Time-Resolved Spectroscopy David Rais, Miroslav Menšík, Bartosz Paruzel, Petr Toman, and Jiri Pfleger J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.8b07395 • Publication Date (Web): 18 Sep 2018 Downloaded from http://pubs.acs.org on September 27, 2018

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The Journal of Physical Chemistry

Concept of the Time-Dependent Diffusion Coefficient of Polarons in Organic Semiconductors and Its Determination from Time-Resolved Spectroscopy

David Rais, Miroslav Menšík,* Bartosz Paruzel, Petr Toman, Jiří Pfleger Institute of Macromolecular Chemistry, AS CR, v.v.i., Heyrovského nám. 2, 162 06 Prague, Czech Republic. *Corresponding author’s email: [email protected]

Abstract The population of photogenerated species in organic semiconductors may decay due to their mutual annihilation upon collisions during their diffusive motion. The standard kinetic models for the population decay, n(t), assume a time-invariant diffusion coefficient, i.e. D(t) ≡ const. This leads to a failure in predicting the experimentally observed temporal evolution of

photogenerated species if it asymptotically approaches a power-law decay n(t) ∼ t-x, with x < 0.5.

We have used a concept of the time-dependent diffusion coefficient and developed a novel mathematical method of its determination from decay collision rates obtained by transient optical absorption spectroscopy. We tested the applicability of this method on the interpretation of data of the decay of polaron population obtained experimentally by time-resolved transient absorption measurements on thin films of regioregular poly(3-hexylthiophene), where we recently reported a power law asymptote with x = 0.24. While we do not assume any microscopic origin of the time variance of D(t), we argue that as the charge-carrier trapping states occupancy drops with

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decaying charge concentration, the carrier motion slows down. This argument is supported by a recent report on a molecular-scale model taking into account significant local anisotropy present in linear conjugated polymers. Our concept can be applied for the description of the evolution of species, like polarons or excitons, in various organic materials, provided their decay kinetics is controlled by a mutual annihilation during 1- or 3-dimensional diffusion.

Introduction Due to an extensive development of time-resolved spectroscopic methods for organic and hybrid systems in recent years, a determination of time-resolved dependences of charge carriers mobility,1–6 electron-hole collision rate,7 electron-hole separation distance8 or exciton-exciton collision rate9 was made possible. It is almost a standard rule that these dependences are very far from simple exponential decay, but rather close to power-law decay characteristics.

The power-law decay of population, n(t) ∼ t-x, indicates mutual annihilation reactions between the excited species. When the annihilation occurs via collisions between the species during their diffusive motion, the assumption of a time-invariant diffusion coefficient always yields the

power-law asymptotic behavior,10 with ≥ 0.5. However, in many cases11–13 the resulting

experimental power-law asymptote has < 0.5 and thus it becomes difficult to interpret exactly experimental data within the collision model with the time-invariant diffusion coefficient.

Recently, the power-law decay of ca ~ . was found for the kinetics of excitons11 and of inter-

chain polaron pairs (PPs)12 in P3HT. In the latter case, it pointed to the bi-molecular collision process of the inter-chain PPs with the 1-d diffusion controlled kinetics. We argued14 that the decay ~ . can be explained assuming a slightly decreasing diffusion coefficient of PPs in

time. Our model of the decay of PPs is significantly different from dissociation model based on the solution to the Smoluchowski equation,15 which leads to the time-dependence ~ .. Even 2 ACS Paragon Plus Environment

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slower decay ~ . was measured for the population of the polarons (P) in regioregular poly(3-

hexyl-thiophene) (P3HT),12 indicating thus a stronger decrease of the diffusion coefficient of

polarons. In this paper we propose the concept of the time-variable diffusion coefficient16 for the theoretical treatment of problems regarding the experimentally determined power-law decay with

exponent < 0.5. Correlation between the time-dependent diffusion coefficients (or hopping rates) and the microscopic structure of the material can be understood on the basis of following arguments: The time evolution of the density matrix of photoexcited system is described on the microscopic level by the Liouville equation,17 however, this equation contains complete information on the relaxation processes within complete electronic and vibration manifold. In experiments with time-resolution longer than periods of vibrational oscillations the vibrational degrees of freedom (DOF) are projected out. In the same way, signals are collected from significantly larger space segments than mean distances between monomer units. Both the space and vibrational averaging correspond to projecting information on state-subspace controlled by time-convolution (time-dependent memory) in the Nakajima-Zwanzig equation.18,19 Equivalently, the system evolution of the projected state-subspace can be reformulated in the Tokuyama-Mori time-local approach,20 where the time-dependence of the memory kernel is removed and the evolution kinetics becomes time-local. However, for the latter, the time-local equations include time-dependent rate constants with relaxation time corresponding to internal dephasing processes (cf. Čápek21). In the same way, the hot local temperature on which charge mobility is dependent22,23 relaxes in time and the charge mobility (diffusion coefficient) slows down accordingly. For systems in which charges are transferred between orbitals of adjacent molecular groups (as, e.g., in the case of polaron movement or charge-transfer (CT) kinetics), the Green’s function technique24 is more convenient than formulation of the description of the system kinetics 3 ACS Paragon Plus Environment


within the Liouville equation approach. Green’s functions enable direct formulation of the theoretical description in terms of fermionic creation and annihilation operators corresponding to particular orbitals. For such interacting systems the higher-order correlation terms can be decomposed utilizing the so-called Wick’s theorem and kinetic master equations become nonlinear in occupation probabilities. This approach was used for calculations of the internal relaxation rate in quantum dots (QD),25 power-law luminescence upconversion,26 and inter-QD electron transfer rate.27,28 It has been also shown recently that experimental data of the timeresolved decay of polaron pairs (PPs) in 10 nm thin films of after photoexcitation to the singlet state can be explained by those non-linear kinetic master equations taking into account up and down conversion processes between inter-chain PPs and polarons on the picosecond time scale.29 The “up/down” electron transfer rates between respective orbitals are proportional to the product ( )(1 − ( )), where n1 and n2 denote instant population of the respective orbitals.

For conjugated polymers like, e.g., P3HT, the molecular scale approach for the molecular scale mobility calculation has been recently developed,30 where the inter-chain charge transfer in the thermalized regime can be described by the following process: In the first step charges are delocalized over a polymer segment, in the next step they thermalize and, finally, they hop to the

adjacent chain with the charge transfer rate proportional to = ∑∈,∈ ( )(1 −

!)" . Here, the indexes # and ' denote the delocalized orbitals on respective chains A and

B, while " describe the charge transfer rates between particular orbitals, which are controlled

by respective inter-chain transfer integrals. This model was also used for the description of the inter-chain charge transfer in polyacetylene31 and in P3HT with acceptors.32 Kinetic rate constants related to the charge transfer between respective chains thus become dependent on the total charge populations on these chains, which also results in the dependence of the mobility of


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charges on their concentration, due to the varying occupation of trapping energy levels within the density of states (DOS). The latter has been confirmed by manifold studies, either assuming Gaussian disorder model,33–42 or introducing a novel molecular-scale approach.30–32 If all effects that include projecting out both spatial and vibrational DOF as well as the presence of traps in the DOS were rigorously taken into account and the Pauli-like master equations for the charge transport between polymer segments were built-up, the resulting kinetics rates would be explicitly dependent on time and on the “mean volume charge concentration” due to the occupation of traps. Moreover, the mean volume charge concentration itself can be dependent on time if a charge recombination is involved. When the Pauli-like equations are rewritten from its discrete limit to the continuous one the resulting Smoluchowski equation will contain generally time-dependent mobility (or diffusion coefficient). It is the purpose of this article to develop a method for resolution of the kinetics of polarons by analysis data of measured transient absorption spectra. We show that this kinetics can be explained by introducing time-dependent diffusion coefficients of polarons, which can be directly obtained from the experimental data in a similar way as it was recently done for PPs.14 For this purpose, we outline a general mathematical concept for the reconstruction of the timedependent diffusion coefficient from experimentally determined time-dependent collision rate constants. This concept can be applied for many similar diffusion-related problems and we proved it on the example of previously published experimental data acquired on P3HT thin films.

Results The transient absorption data analyzed in this paper were taken from our earlier work.12 In that paper, we reported transient absorption (TA) spectra of the thin films of P3HT in the probe wavelengths ranging 440–1300 nm observed in the films after excitation with linearly polarized, 5 ACS Paragon Plus Environment


ultrafast light pulses of 550 nm central wavelength at relatively high pulse fluencies; for more detail, see the “Experimental section” in the Supporting Information. The negative TA signal was found at a broad probe wavelengths range ca 470–620 nm and assigned to a ground state bleaching (GSB) phenomenon, due to its similarity with the steady-state absorbance spectrum. Other three positive TA bands, corresponding to the excited state absorption (ESA) signals, were found with peaks at probe wavelengths 650 nm, 950 nm and 1290 nm, and assigned to PP, P, and singlet exciton (SE), respectively, providing similar values as measurements of the bulk solid reported by other research groups.43,44 It was found that SE population is created practically immediately with the photoexcitation pulse followed by their rapid deexcitation. The PPs were created with ca. 200 fs delay, their decay was correlated with formation of P states29. We briefly summarize experimental observations in the section “Evolution of excited species in P3HT” in the Supplementary Information. In this paper, we will concentrate on the evolution of population of photoexcited polarons.

Evolution of polarons population The spectral decomposition of the near infra-red probe region of TA spectral data was used12 to obtain experimental evolution of polaron population (free charge carriers). In Figure 1, we show that in the delay time range between 1 ps and 1 ns the experimental evolution can be fitted well

with a power-law decay (~ . ). Here, we should stress that fitting the experimental data with the kinetic model based on simultaneous polarons generation from dissociation of polaron pairs and their annihilation controlled by the time-invariant diffusion coefficient was not possible (See section “Alternative model with constant diffusion coefficient” in the Supplementary information.).


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Normalized polaron population, P(t)

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100

10-1

100

experiment fit ~ t - 0.24 101

102

103

Time, t (ps) Figure 1: Normalized polaron population decay observed in a P3HT thin film obtained from the spectral decomposition of NIR-probe transient absorption experiment, data taken from Ref.12

The kinetic model derivation In our model, the bi-molecular mutual annihilation of polarons can be described by the process:

P ) + P → 2- . This occurs, when two polarons of opposite charges recombine during their diffusion throughout the solid, yielding no successor excited state in the material. The concentration of polarons in time, n(t), is then driven by the second-order kinetic equation:

./(0) .0

= −γ( ) ( )

(1)

in which γ( ) is the time-dependent collision rate of the polarons. In our model, we also neglect

spontaneous recombination of the space correlated pairs P ) + P with distance between mutual 7 ACS Paragon Plus Environment


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charges longer than the overlap of their orbitals (e.g. charges separation is longer than two interchain distances). Such charges are already detected by the probe pulse as “free”. They are still close to each other as they were created from dissociation of the same exciton but their annihilation would contribute to the single-molecular decay process. Among all diffusion controlled processes of polaron decay which involve either single-molecular or bi-molecular collision with a time independent polaron diffusion coefficient the slowest one is the 1-d bimolecular process with the long-time asymptote ~ .. However, our experimental data show a

much slower polaron decay ~ . , which indicates that the diffusion coefficient could decrease with time. On the other hand, single-molecular processes, as the above mentioned recombination of correlated polarons or trap assisted recombination and impurity ionization, would contribute to the exponential-like decay, which is faster than .. The goal of this section is to derive the time-dependent collision rate γ( ) of the diffusion Introducing time-dependent diffusion coefficient

controlled collision process under assumption of time-dependent diffusion coefficient 1( ). We also determine the inverse relation between time-dependent diffusion coefficient 1( ) and the

collision rate γ( ), which can be determined from experiment. We will follow similar approach,

which was recently utilized for the analysis of the time-dependent diffusion coefficient of polaron

pairs14. For the time-dependent diffusion coefficient 1( ), the diffusion equation, which controls kinetics of species concentration spatial distribution, (2, 334 ), in k-dimensional space reads as .

.0

(2, 334 ) = 1( )∆k (2, 334 ).

(2)

Eq. (1) can be easily rewritten to the form


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.

7(0) .0

(2, 334 ) = ∆k (24, ).

(3)

Introducing a new variable

8 ≡ : 1(;)d; 0

(4)

Eq. (3) is easily rewritten to an equivalent form

.

.
?@

A

B71d C0

,

(6)

and =Dd ( ) = 8F13d H(1 +

I

√C0

) ,

(7)

respectively. In (Eqs, 6, 7) we replace " → 8" , "D→1" and multiply them by 1( ), because the

differential time increments in real time satisfies the equation d = d8/1( ). Finally, we can

write the collision rates, =d ( ) and =3d ( ) for the time-dependent diffusion coefficients =d ( ) =

71d (0) >?@

AC : O 7 @

B

1d (N)dN

=Dd ( ) = 8F13d ( )H(1 +

≡ >?

O

I

@

.

.0

AD :@ 71d(N)dN,

AC : 73d (N)dN @

O

C

(8)

) ≡ 8FH .0 (: 13d (;)d; + H A .

0

O

:@ 73d (N)dN C

)

(9)

for 1d (D1d(t)) and 3d (D3d(t)) cases, respectively.

In Eqs. (8-9) P is the 1d-lattice coefficient, Q is the 3d molecular density, and H corresponds to 3d critical recombination distance. We also note straightforwardly from Eq. 2 that mean square displacement 〈|24| ( )〉 in k-dimensional space can be expressed as

〈|24| ( )〉 ≡ 2k8 = 2U : 1(;)d;. 0

(10)


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The time-dependent collision rates =d ( ) and =3d ( ) can be obtained from experimental data for arbitrary systems, where the species population decay is controlled by the collision events. Once the time-dependent collision rates =d ( ) and =3d ( ) have been experimentally determined, solving Eqs. (8-9) we find directly for arbitrary kinetic processes in either 1-d or 3-d cases:

〈|24| ( )〉d = F

>V ?@

11d ( ) = F

W

X: =1d (;)d;Y ,

(11)

=d ( ) : =1d (;) d; ,

(12)

W

>V ?@

0

0

and

〈|24| ( )〉Dd = 6

13d ( ) =

BCI

IV C

(A1 − I[ : =3d (;)d; − 1) ,

\1 −

0

O ] A [ :@ `3d (N)dN ^_

a =3d ( ).

(13)

(14)

Eqs. (12, 14) represent the inversion solution to Eqs. (8, 9), relating the time-dependent diffusion coefficients to the collision kinetic rates. We note that the diffusion coefficient 1( ) corresponds to one species, while the diffusion coefficient of the relative motion of these species is 21( ).



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Eqs. (11-14) express the diffusion coefficient 1( ) of a stochastic motion of arbitrary species

Decay process with time-dependent diffusion coefficient

with concentration ( ) obtained from its annihilation rate =( ). However, such decay rate may

take a general functional dependence =( ) = ={ ( ), ( ), … } stemming from all coupled

species in the collision event. At this point, we should mention that during localization of photoexcited species their oscillator strength can vary in time and, hence, temporal changes in the oscillator strength of the probe could be possibly erroneously misinterpreted as population changes. To avoid this artefact a proper spectral decomposition of the experimentally obtained absorption spectra must be performed, so that for the photoexcited manifold { ( ), ( ), … } their respective absorption coefficients {e (f), e (f), … } will not vary in time.

Let’s assume for simplicity that the decay process of a particular excited state species is not coupled with other species and that the collision process is proportional to an arbitrary functional

dependence g{( )} on the species concentration ( ). For the collision decay on static impurities it takes linear form with respect to ( ) and for the mutual collision process it takes

the quadratic form with respect to ( ). The population decay rate is then controlled by the

following equation

d/(0) d0

= −=( )g{( )}

(15)

and for Eqs. (9-12) we find for the 1d case

〈|24| ( )〉d = F

>V ?@ B

(:/( ) i(h)) , /(0) dh

(16)


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11d ( ) = F

>V ?@ B

d/(0)

i{/(0)} d0

/(0) dh

:/( ) i(h) .

(17)

Eqs. (16, 17) are expressed for any general recombination process controlled by Eq. (15). Then, in the case of the “bi-molecular” collision we replace the factor 8 by 16 in the denominator. Similarly for the 3d case of the relative motion we get

〈|24| ( )〉Dd = 6 13d ( ) =

BCI

IV C

(A1 + I[ :/( ) i(h) − 1) ,

(1 −

/(0) dh

m(O) dk ] A) [ :m(@) l(k) ^j

) i{/(0)}

d/(0) d0

(18)

.

(19)

For more complex systems including mutual annihilations between multiple types of species { ( ), ( ), … } we can determine either analytically or numerically collision rates from profiles

of their populations and their derivatives in dependence on particular coupling scheme (cf. Supplementary information of Ref.7) and use expressions Eqs. (16-19). They can be also applied

for coupling of more species { ( ), ( ), … }, when diffusing species ( ) interacts, e.g., with the species ( ) and then, in performing the numerical integration :/ ]( ) i(h / (0) V

dh]

] ,hV )

for the value

n we set n ≡ ( = ;) at the time ;, for which (;) = n . A special case is the decay process controlled by the species collision on static impurities (reaction centers), so called singlemolecular process, which was also investigated for PPs decay.14 However, we neglect this process as only polarons possessing mutually opposite charges can annihilate by their collisions,



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and we do not assume significant content of charge recombination centers in the neutral samples of thin film P3HT. The decay of species population ( ) controlled by the bi-molecular collision (like collision of Bi-molecular collision processes

excitons, polaron pairs or oppositely charged polarons) can be related with the time-dependent diffusion coefficient 1( ) (equal to half of that for the relative motion of two identical species involved in the collision process) in the following way:

a) 1d case: d/(0) d0

71d (0)

= −AC >? B

@A O :@ 7]d (N)dN

( ).

(20)

providing

C(>? )V @

〈|24| ( )〉1d ≡ 2 : 1o (;)d; = p − /( )q /(0) 0

11d ( ) =

C(>?@ )V W

W

,

() d/(0)

{/(0) − /( )} /(0)V

d0

.

(21)

(22)

Inserting experimental data of species population decay obtained from time-resolved spectroscopy to the right hand side of Eqs. (21-22) we get direct information on time dependence of mean square displacement 〈|24| ( )〉1d and diffusion coefficient 11d ( ). For the power-law

population decay ( ) ~ we find 11d ( ) ~ { − 1} → . Only for # =


the

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diffusion coefficient is constant in time. Similarly, for the mean square displacement we find 〈|24| ( )〉1d → and only for # =

it increases linearly with time.

b) 3d case: d/(0) d0

= −8FH \21Dd ( ) +

I7[d (0)

O AC : 7[d (N)dN @

a ( )

(23)

〈|24| ( )〉3d ≡ 6 : 1Dd (;)d; = 6{A X − /( ) + 4H D Y − BCI /(0) 0

1( )3d = {1 − WCI

I

√C

}

() d/(0)

} /(0)V

] m(@) A X Y) ^m(@)j[ m(O)

d0

(24)

(25)

Similarly as in the 1d case, the mean square displacement 〈|24| ( )〉 and diffusion coefficient 1( )

in Eqs. (24 - 25), can be obtained from a time-resolved measurement of population decay ( ). However, contrary to the previous case these kinetics are also determined by the dimensionless

parameter 4(0)H D , which scales the mean effective annihilation volume with respect to the

initial population concentration. We easily find that formally in the limit 4(0)H D → ∞ (i.e. infinitely large “annihilation volume” with respect to the species concentration) expressions valid for 3d-case change to the 1d ones. For real initial species concentrations we expect rather opposite limit 4(0)H D < 1 or even 4(0)H D ≪ 1. Namely, for the polaron recombination radius

b corresponding to the mean monomer distance the limiting value 4(0)H D = 2 corresponds to

50 % occupation of monomers by P ) and 50 % occupation for P polarons. For the power-law

population decay ( ) ~ in the limit of asymptotically long times we also find for the diffusion coefficient 1( ) ~ and for the mean square displacement 〈|24| ( )〉 → . 15 ACS Paragon Plus Environment


Discussion Eqs. (22) and (25) provide a recipe how to reconstruct the time dependence of the diffusion coefficient of polarons (or excitons) in a material with a particular dimensionality of the molecular structure from experimentally determined time dependence of polaron (exciton) concentration. It is derived for the case of the population decay controlled by the bi-molecular collisions but without knowing the physical origin of such dependence. Here, the physical origin can be found in several processes: a) Increasing effective mass of polarons or excitons in time due to increasing dressing of the moving charge or excitation with phonons. This can occur in the case when the local charge (exciton) coupling becomes more effective when its movement slows down. b) Distribution of local structural and energetic disorder30 is not homogenous in space, but domains with lower degree of disorder and, consequently, a faster charge diffusion are separated by spare regions of higher disorder (and thus a slower charge mobility). This will influence the charge or exciton diffusion coefficient, which can decrease during movement over larger distances. The extracted decay of diffusion coefficient with time is then indirect effect of the time dependent mean free path. c) Deep trap state filling effects in disordered systems can result in an increase of the charge mobility with increasing charge concentration in the range 0.001-0.1 per monomer unit.30 As the charge concentration decreases in time, the diffusion coefficient will decrease in time, too. In Figures 2-4, we show the respective correlation graphs, in which the diffusion coefficient values were calculated using Eqs. (22) and (25). As the input data for the timedependent occupation probabilities inserted into the right hand side of these equations, we used the power-law fit ~ . , which reproduces the experimental data well in the delay time interval between 1 ps and 1 ns (see Figure 2). This substitution was used to avoid amplification of the 16 ACS Paragon Plus Environment

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data fluctuation when calculating the first derivative of the time course of the polaron concentration,

du(0) d0

, according the Eqs. (22, 25).

In Figure 2, we show the resulting evolutions of normalized polaron diffusion coefficient

1( ) for the 1d and 3d models, respectively. In each of the 3d cases we assumed different value of 4v(0)H D (mean effective annihilation volume with respect to the initial polaron population

concentration). It is evident that the “large recombination volume” limit provides results closer to the 1d case. This case is however not very realistic as the value 4v(0)H D = 2 would require that

the recombination radius of polarons, corresponding to the mean monomer distances of P3HT, would be comparable to the mean initial separation distance between charges. Other ratios of mean effective annihilation volume to the initial polaron concentration showed marked decrease of polaron diffusion coefficient in short times, which drops down at least by two orders of magnitude between 1 ps and 1 ns. On the other hand, the 1d model showed much slower decay of only one order of magnitude at the same time scale.


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Diffusion coefficient, D(t)/D(0)


1d 3d, 3d, 3d,

100

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4P(0)b3 = 2 4P(0)b3 = 0.2 4P(0)b3 = 0.02

10-1

10-2

10-3 100

101

102 Time, t (ps)

103

Figure 2: Normalized values of time dependent polaron diffusion coefficients for 1d and 3d models. For the latter case the relative values of initial effective recombination volume 4v(0)H D = 2 (dash), 4v(0)H D = 0.2 (dotted), and 4v(0)H D = 0.02 (dash-dotted) are shown.

In Figure 3, we show the correlation of the 1d diffusion coefficient 1( ) with mean polaron

diffusion distance 2( ) ≡ w〈|24| ( )〉 (cf. Eq. 16). For the latter we show values in arbitrary units as from the experimental data we do not know directly the absolute value of polaron

concentration v( ), which are required in Eqs. (21–22). We could, however, estimate indirectly the concentration of polaron density at 1 ns to be ca 1.2 % with respect to the number of monomer units, knowing that the initial excitation population was ca 20 % of monomer units,12 GSB at 1 ns is ca 16 times lower than the initial GSB, and that its time course shows almost identical decay profile as v( ) (see Figure S3 in the Supplementary Information). One can see in 18 ACS Paragon Plus Environment

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Figure 3 that for shortest diffusion distances the diffusion coefficient only slightly decreases upon increasing the distance by a factor of 2, while for large polaron travelling distance the diffusion coefficient decreases quite significantly. In structurally inhomogeneous materials it could be explained by the fact that on short distances the polaron diffuses within some ordered domains or conjugation segments, while for large distances the polaron has to overcome some energy barriers between these domains.

1d diffusion coefficient, D1d(t)/D1d(0)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


100

10-1

10-2

100 101 Mean 1d polaron diffusion distance r(t) (arb.u.)

Figure 3: Correlation relation of the normalized 1d diffusion coefficient 11d ( ) with the mean 1d polaron diffusion distance r(t).

In Figure 4, we show correlation of the normalized diffusion coefficient with respect to the hole concentration for 1d or 3d model, respectively, considering various recombination radii. The normalized polaron concentration in Figure 1 was related to the hole concentration per monomer units due to the following arguments: For time t = 1 ns we find in Figure 1 the polaron 19 ACS Paragon Plus Environment


concentration v ≅ 11 % of initial polaron concentration. Next, we also assume that the concentration of

v ) and v polarons are equal since they are formed in pairs upon

photoexcitation of the neutral system. The maximum value of diffusion coefficient 1( ) in Figure 4 is achieved for the hole concentration of about 4-5 % per monomer unit.

100 Diffusion coefficient, D(t)/D(0)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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10-1

10-2 1d 3d, 3d, 3d,

10-3

10-4

4P(0)b3 = 2 4P(0)b3 = 0.2 4P(0)b3 = 0.02

10-2

10-1

Hole concentration per monomer unit

Figure 4: Correlation of normalized diffusion coefficient with respect to hole concentration for 1d and 3d model, respectively, for various recombination radii. For the latter case the relative values of initial effective recombination volume 4v(0)H D = 2 (dash), 4v(0)H D = 0.2 (dotted), and 4v(0)H D = 0.02 (dash-dotted) are assumed.

The dependence of the diffusion constant on hole concentration shown in Figure 4 excellently corresponds to the mobility dependence on charge concentration obtained from a molecular scale


Page 21 of 35

modelling and shown in Figure 5. It has been shown for both crystalline and amorphous phases that the initial increase of diffusion coefficient (or mobility) corresponds to the traps filling, which saturates at high hole concentrations v ≈ 2 − 5 % holes per monomer unit and then

decreases with increasing hole concentration due to the lack of available vacant transport energy states.

101 100

Mobility, µ (cm2/(Vs))

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


10-1 10-2

Crystalline phase

10-3 10-4 10-5 10-6

Amorphous phase

10-7 10-8 10-5

10-4

10-3

10-2

10-1

100

Hole concentration per monomer unit

Figure 5: Mobility dependences on charge carrier concentration in P3HT in crystalline and amorphous state, respectively, calculated from molecular-scale modelling (data taken from Ref.30) for various energy disorder σ: a) σ = 0, solid line, b) σ = 0.05 eV, dashed line, c) σ = 0.01 eV, dotted line, d) σ = 0.015 eV, dash-dotted line. Red and black curves are for the crystalline and amorphous phases, respectively.



Conclusions

We have explained the very slow decay of polaron concentration ~ . in 10 nm thin films of regioregular poly(3-hexyl-thiophene) observed by transient absorption optical spectroscopy at time interval 1 ps – 1 ns by introducing the concept of time-dependent diffusion coefficient. We have developed a novel approach for the analysis of the transient absorption spectra that allows us to extract the time-dependent diffusion coefficient from the experimental data of the timedependent collision controlled decay rates of photoexcited species concentration. We also showed that correlation between the time-dependent diffusion coefficient and the decaying polaron concentration corresponds to the dependence of the mobility charge carriers on their concentration, calculated recently for the energetically and structurally disordered P3HT using a molecular-scale approach.30 Although our method was illustrated for the particular case of bimolecular collision scheme it can be applied also for other arbitrary coupled populations of particular species (polarons or excitons) in any organic material provided their decay kinetics is controlled by mutual annihilation during 1- or 3-dimensional diffusion.

Acknowledgement This work was supported by the Czech Science Foundation (Project No. 17-03984S), and the Ministry of Education, Youth and Sports of the Czech Republic (Project COST LTC17029COST Action MP1406).

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For Table of Contents Only



Graphical abstract, i.e., Table of contents 133x90mm (150 x 150 DPI)

ACS Paragon Plus Environment

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Figure1: Normalized polaron population decay observed in a P3HT thin film obtained from the spectral decomposition of NIR-probe transient absorption experiment, data taken from Ref. 12. 254x190mm (96 x 96 DPI)



Figure 2: Normalized values of time dependent polaron diffusion coefficients for 1d and 3d models. For the latter case the relative values of initial effective recombination volume 4P(0)b3=2 (dash), 4P(0)b3=0.2 (dotted), and 4P(0)b3=0.02 (dash-dotted) are shown. 254x190mm (96 x 96 DPI)


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Figure 3: Correlation relation of the normalized 1d diffusion coefficient D1d (t) with the mean 1d polaron diffusion distance r(t). 254x190mm (96 x 96 DPI)



Figure 4: Correlation of normalized diffusion coefficient with respect to hole concentration for 1d and 3d model, respectively, for various recombination radii. For the latter case the relative values of initial effective recombination volume 4P(0)b3=2 (dash), 4P(0)b3=0.2 (dotted), and 4P(0)b3=0.02 (dash-dotted) are assumed. 254x190mm (96 x 96 DPI)


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Figure 5: Mobility dependences on charge carrier concentration in P3HT in crystalline and amorphous state, respectively, calculated from molecular-scale modelling (data taken from Ref.30) for various energy disorder σ: a) σ = 0, solid line, b) σ = 0.05 eV, dashed line, c) σ = 0.01 eV, dotted line, d) σ = 0.015 eV, dashdotted line. Red and black curves are for the crystalline and amorphous phases, respectively. 254x190mm (96 x 96 DPI)


Concept of the Time-Dependent Diffusion Coefficient of Polarons in

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