Power-law density of states in organic solar cells revealed by the open-circuit voltage dependence of the ideality factor Maria Saladina1Christopher W opke1Clemens G ohler1Ivan Ramirez2Olga Gerdes2Chao Liu3 4Ning Li3 4 5

2025-05-02 0 0 640.39KB 12 页 10玖币

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Power-law density of states in organic solar cells revealed by the open-circuit voltage

dependence of the ideality factor

Maria Saladina,1, ∗Christopher W¨opke,1Clemens G¨ohler,1Ivan Ramirez,2Olga Gerdes,2Chao Liu,3, 4 Ning Li,3, 4, 5

Thomas Heum¨uller,3, 4 Christoph J. Brabec,3, 4 Karsten Walzer,2Martin Pfeiﬀer,2and Carsten Deibel1, †

1Institut f¨ur Physik, Technische Universit¨at Chemnitz, 09126 Chemnitz, Germany

2Heliatek GmbH, 01139 Dresden, Germany

3Institute of Materials for Electronics and Energy Technology (i-MEET),

Friedrich-Alexander-Universit¨at Erlangen-N¨urnberg, 91054 Erlangen, Germany

4Helmholtz Institute Erlangen-N¨urnberg for Renewable Energy (HI ERN), 91058 Erlangen, Germany

5State Key Laboratory of Luminescent Materials and Devices,

Institute of Polymer Optoelectronic Materials and Devices, School of Materials Science and Engineering,

South China University of Technology, 510640 Guangzhou, China

The density of states (DOS) is fundamentally important for understanding physical processes in

organic disordered semiconductors, yet hard to determine experimentally. We evaluated the DOS by

considering recombination via tail states and using the temperature and open-circuit voltage (Voc )

dependence of the ideality factor in organic solar cells. By performing Suns-Voc measurements, we

ﬁnd that gaussian and exponential distributions describe the DOS only at a given quasi-Fermi level

splitting. The DOS width increases linearly with the DOS depth, revealing the power-law DOS in

these materials.

The dominant recombination mechanism in a solar

cell is intimately related to the ideality factor.1–3 For

inorganic semiconductors, the closer the ideality factor

gets to 2, the more dominant the share of trap-assisted

recombination.4This connection is more complex for or-

ganic materials used in the state-of-the-art solar cells due

to the energetic disorder inherent to these systems, giving

rise to wave function localisation. Consequently, charge

carrier transport and recombination in these disordered

materials strongly dependent on the energetic distribu-

tion of localised states.5–9 If the density of states (DOS)

can be approximated by a gaussian, the ideality fac-

tor becomes unity and independent of temperature.9,10

However, temperature-independent ideality factors equal

to 1 are yet to be reported for organic donor–acceptor

systems,11–14 implying that the DOS is more complicated

in these materials.15–17

In this Letter, we seek to unravel the real shape of

DOS in a set of solar cells based on organic semicon-

ductor blends. To achieve this objective, we deter-

mine ideality factors from temperature-dependent Suns-

Voc measurements. We connect the results to theoretical

predictions by the multiple-trapping-and-release (MTR)

model7,8,18,19 using diﬀerent combinations of the gaus-

sian and exponential DOS functions20–25 to describe the

energetic state distribution of electrons and holes. De-

pending on the shape of DOS and the dominant recom-

bination mechanism, the temperature dependence of the

ideality factor diﬀers,9which allows us to assign speciﬁc

recombination models to the investigated systems. We

ﬁnd that the width of the DOS distribution in these or-

ganic solar cells depends on the energetic position in the

∗email: maria.saladina@physik.tu-chemnitz.de

†email: deibel@physik.tu-chemnitz.de

DOS, resulting in a power-law distribution of localised

states.

Current–voltage characteristics of a solar cell are usu-

ally approximated by the diode equation.26 The recom-

bination rate Renters the diode equation via recombina-

tion current density jrec, which at the open-circuit con-

ditions takes the form27

jrec =eZL

R(x)dx ≈eLR

=j0·exp eVoc

nidkBT.

(1)

Here Lstands for the active layer thickness, j0the dark

saturation current density, nid the ideality factor, Voc

the open-circuit voltage, ethe elementary charge, kB

the Boltzmann constant, and Tthe temperature. In the

framework of the extended Onsager model,28–30 recombi-

nation takes place via the formation of a charge-transfer

(CT) state, when charge carriers are within a certain dis-

tance of one another. Consequently, for recombination to

occur at the donor–acceptor interface, at least one of the

charge carriers needs to be mobile, and recombination of

trapped charge carriers with each other is excluded.31,32

Then, the recombination rate is expressed as

R=kr,npncpc+kr,nncpt+kr,pntpc,(2)

where krstands for the recombination prefactor, and the

subscript denotes which of the charge carriers, electrons

nor holes p, are mobile. We use diﬀerent prefactors to

highlight the dependence of kron the combined mobility

of charge carriers. The ﬁnite escape probability from

CT back to the separated state, if present, is included

in krand reduces it compared to the Langevin prefactor

kL=e(µn+µp)/ε.33–35 Additionally, krcontains the

eﬀects of active layer morphology that cause its further

deviation from kL.36–38

arXiv:2210.11208v1 [physics.app-ph] 20 Oct 2022

In organic disordered materials, the majority of lo-

calised states in the DOS lie below the transport

energy,39 and act as traps, which capture mobile charges.

Trapped charge carriers can be thermally released and

contribute to photoconductivity. During this process

of multiple-trapping-and-release,7,8,18,19 some share of

charge carriers recombines and is lost to the photocur-

rent. In the MTR model, the fraction of the mo-

bile charge carrier density is expressed through parame-

ter θ, the trapping factor, which depends on the DOS

distribution.7,19,40 The density of mobile and trapped

charge carriers is expressed as nc=θn and nt=

(1 −θ)n, respectively, with θ < 1. As the eﬀective den-

sity of trap states is much larger than the charge carrier

density, most relaxed charge carriers will populate energy

sites in the DOS tail.41 Thus, ncnt≈nand θ1, in-

ferring that recombination is mainly trap-mediated and

making the ﬁrst term in Eq. (2) negligible. Using the

above notations, and noting that due to the nature of

photogeneration, n=p, the recombination rate becomes

R≈(kr,nθn+kr,pθp)n2.(3)

The two recombination channels in Eq. (3) are distin-

guished by the type of mobile charge carrier. One of

the channels is dominant if its recombination prefactor

and/or its trapping factor is larger than for the other

channel. Thus, the exact expression of Rdepends on

(i) the physical parameters, e.g. mobility, of the mobile

charge carrier type through the recombination prefactor

kr, (ii) the DOS of this charge carrier type through the

trapping factor θ, and (iii) the DOS of the more abun-

dant type of charge carrier in the dominant recombina-

tion channel through the total charge carrier concentra-

tion n.

Herein, we focus on the most prevalent models used to

approximate the DOS distributions in organic semicon-

ductors – the gaussian and exponential DOS,20–25 and

their inﬂuence on the ideality factor. The depth of trap

states, corresponding to the width of the distribution,

depends on the disorder parameter σand the Urbach en-

ergy EU, respectively. The resulting form of Eq. (3) is

deﬁned by four combinations of these DOS distributions.

The ﬁrst two involve electrons and holes being described

by the same DOS distribution, whether gaussian, or ex-

ponential, and, for the sake of simplicity, θn=θp. If,

however, the DOS functions of electrons and holes are

diﬀerent, and θn6=θp, the eﬀective recombination rate

is additionally determined by the type of mobile charge

carrier. For the detailed derivation, the interested reader

is referred to the comprehensive work of Hofacker and

Neher.9Here, we build on a mere fraction of their re-

sults related to the ideality factor and summarise rele-

vant parts of the derivation in the Supplemental Mate-

rial. The ideality factor is obtained by comparing Eq. (1)

to the equations of Rfor the DOS combinations discussed

above (Eqs. (S11) to (S14)).

Without loss of generality, we describe the dominant

recombination channel involving mobile holes recombin-

ing with trapped electrons. The more abundant type

of charge carrier in the recombination channel (nt≈n)

controls the temperature dependence of the ideality fac-

tor, while the mobile charge carrier type (pc=θp·p)

controls the recombination order. If the DOS of elec-

trons is described by an exponential, the ideality factor is

temperature-dependent. When such electrons recombine

with mobile holes from the gaussian DOS, the ideality

factor is independent of σand is expressed as9

nid =EU+kBT

2kBT,(4)

If mobile holes are also represented by the exponential

DOS, the ideality factor is given by9,12,42,43

nid =2EU

EU+kBT.(5)

We will be referring to these models as the mixed DOS

and exponential DOS, respectively. In contrast, if the

DOS of electrons is described by a gaussian, in the low

concentration limit we arrive at nid = 1, independent

of temperature.9,10,44 This is true irrespective of whether

mobile holes come from the gaussian or exponential DOS.

An ideality factor of unity is not observed experimen-

tally in organic semiconductors,11–14 which leads to two

implications. Firstly, in a mixed DOS, the dominant re-

combination channel is the one involving mobile charge

carriers in the gaussian recombining with trapped charges

in the exponential DOS. A gaussian DOS reaches less

deep into the band gap so that θis generally closer to

one than for an exponential DOS. Hence, this channel

will have a larger share of mobile charge carriers lead-

ing, for the same kr, to a larger eﬀective recombination

prefactor than for the other channel. Secondly, the total

distribution of localised states is likely more complicated

than the gaussian for organic materials.

Consequently, in order to shed light on the shape of the

DOS in these systems, our focus should lie on the temper-

ature dependence of the ideality factor, with the models

underlying Eqs. (4) and (5) as the starting point. The

distinct temperature dependence of nid in these expres-

sions allows us to determine the prevailing recombination

mechanism in a solar cell dominated by trap-assisted re-

combination, and the likely form of DOS distribution.

To verify that the DOS can be established through

ideality factors, we chose the well-studied hydrogenated

amorphous silicon solar cell as a reference. We then ex-

pand our investigation to a set of material systems rep-

resentative of typical organic solar cell classes, such as

solution-processed fullerene (P3HT:PC61BM) and non-

fullerene acceptor devices (PM6:Y6), along with ther-

mally evaporated small-molecule solar cells (DCV-V-Fu-

Ind-Fu-V:C60). The details of molecular structure and

device fabrication are given in the Supplemental Mate-

rial.

We employ illumination intensity-dependent Voc mea-

surements to determine ideality factors in the absence

Figure 1. (a) Suns-Voc data (symbols) of a-Si:H solar cell

ﬁtted with Eq. (6) (solid lines). (b) Temperature-dependent

ideality factor nid extracted from the ﬁts in comparison to

Eqs. (4) and (5).

of series and transport resistance.45–47 Figure 1(a) shows

the Suns-Voc data of a-Si:H solar cell between 150 K and

300 K, excluding the regions of low light intensity in-

ﬂuenced by low shunt resistance. Roughly above 1 sun,

Voc becomes limited by the contacts, which is more pro-

nounced at low temperatures. Ideality factors were ex-

tracted from the slope of Φ(Voc) according to13

nid =e

kBTdln Φ

dVoc −1

.(6)

For each temperature, the data can be ﬁtted with a single

slope over two orders of magnitude of light intensities.

We plot the resulting ideality factors against the in-

verse temperature in Figure 1(b). Consistent with van

Berkel et al.,42 we observe the decrease of nid of a-Si:H

solar cell with higher temperature from 1.7 at 150 K to

1.5 at 300 K. The temperature dependence of nid can be

ﬁtted with Eq. (5) and therefore is assigned to the trap-

assisted recombination of charge carriers with the expo-

nential density of states. In the exponential DOS model

nid ∝(EU+kBT)−1, resulting in sublinear dependence

on 1/T . The ﬁt yields the Urbach energy of ≈78 meV, in

agreement with the literature,42,48 which is independent

of temperature and light intensity. A mixed DOS would

lead to a distinctly diﬀerent temperature dependence of

the ideality factor, as nid ∝1/T according to Eq. (4).

We now extend the scope of the study to organic

donor–acceptor systems. Figure 2 shows ideality fac-

tors of P3HT:PC61BM, PM6:Y6 and DCV-V-Fu-Ind-Fu-

V:C60. First, we note that at each temperature nid has

several values corresponding to the local slope of Φ(Voc).

Hence, in contrast to a-Si:H, the ideality factor of the

organic systems we investigate here is light intensity-

dependent, and generally decreases with increasing light

intensity. At ﬁrst, it seems problematic to assign a spe-

ciﬁc recombination model based on the temperature de-

pendence of nid at a certain illumination intensity (cf.

Figure S3). This method does not account for potential

changes of DOS shape with the quasi-Fermi level splitting

(QFLS), which varies with temperature for ﬁxed light in-

tensity.

Instead, we evaluate the data at ﬁxed Voc. The QFLS,

approximated by Voc, samples the combined DOS of elec-

trons and holes at a certain energy. The Urbach energy,

Figure 2. Temperature-dependent ideality factors nid of

(a) P3HT:PC61BM, (b) PM6:Y6 and (c) DCV-V-Fu-Ind-Fu-

V:C60 (symbols). Darker color corresponds to lower Voc,

i.e. deeper subgap energy states. Dashed lines are the cal-

culated nid(EU, T ) according to Eq. (5) for P3HT:PC61BM

and Eq. (4) for PM6:Y6 and DCV-V-Fu-Ind-Fu-V:C60 .EU

increases with the DOS depth for all three systems.

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Power-lawdensityofstatesinorganicsolarcellsrevealedbytheopen-circuitvoltagedependenceoftheidealityfactorMariaSaladina,1,ChristopherWopke,1ClemensGohler,1IvanRamirez,2OlgaGerdes,2ChaoLiu,3,4NingLi,3,4,5ThomasHeumuller,3,4ChristophJ.Brabec,3,4KarstenWalzer,2MartinPfeier,2andCarstenDeibel1,y1Insti...

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Power-law density of states in organic solar cells revealed by the open-circuit voltage dependence of the ideality factor Maria Saladina1Christopher W opke1Clemens G ohler1Ivan Ramirez2Olga Gerdes2Chao Liu3 4Ning Li3 4 5.pdf

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Power-law density of states in organic solar cells revealed by the open-circuit voltage dependence of the ideality factor Maria Saladina1Christopher W opke1Clemens G ohler1Ivan Ramirez2Olga Gerdes2Chao Liu3 4Ning Li3 4 5

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