Supplementary Material
A dynamic Monte Carlo study of anomalous current voltage behaviour in organic solar
cells
Krishna Feron,1,
2, a)
Xiaojing Zhou,1 Warwick J. Belcher,1 Christopher J. Fell,1,2 Paul C.
Dastoor1
1
Centre for Organic Electronics, University of Newcastle, Callaghan, NSW 2308, Australia.
2
CSIRO Energy Technology, Newcastle, NSW 2300, Australia
a) corresponding author. Electronic mail: Krishna.Feron@csiro.au
S-A. Generating 3D morphologies
A widely accepted cellular automate method is used to generate 3D morphologies1. This
algorithm does not necessarily simulate the phase separation process during thermal
annealing process, but is simply used to generate a morphology that is representative of a real
bulk heterojunction (BHJ) structure. An initial BHJ morphology was generated by occupying
a 3D Cartesian lattice with moiety identifiers (zero for PCBM and one for P3HT),
representing either P3HT or PCBM. These numbers were assigned to random locations with
equal probability to ensure 1:1 P3HT:PCBM ratio. The feature size, which is defined as 3 x
volume / interface area, of the initial structure is increased by swapping neighbouring sites.
Swaps are attempted and accepted if the energy of the system decreases or with a probability
defined by the change in the system’s energy. Energy of site i is calculated using
𝐽
𝐸𝑖 = − ∑ (𝛿𝑠𝑖 ,𝑠𝑗 − 1)
2
𝑗
where s is the moiety identifier, δ the Kronecker delta and J the exchange interaction, which
was chosen as 1.The summation over j indicates the summation over all nearest neighbours.
Lowering the system energy results an increase in feature size. Swapping of single sites is
1
good for generating BHJ structures with small feature size, but becomes a very slow
computational approach for generating structures with large feature size. Hence, to generate
systems with large feature sizes, a cluster algorithm is used. A modified version of Wolff’s
algorithm2 is implemented where the lattice is divided in clusters. A cluster is formed by first
randomly choosing a starting point. If neighbouring sites have the same moiety identifier they
are added to the cluster with probability p. In this manner the entire lattice is divided in
clusters followed by swapping clusters of equal size but different moiety identifiers to
preserve the 1:1 ratio. Since many sites are swapped in one go, the algorithm is significantly
faster. Another advantage of the cluster algorithm is that small islands (one or two voxels) are
quickly avoided. The single site swapping algorithm sometimes produces these small islands,
which is non-physical since they do not correspond to photoluminescence quenching
measurements and polymer are longer than 1 voxel3. Fig. S1 shows the evolution of a BHJ
morphology during the swapping algorithm. The morphology on the right of Fig. S1 had a
feature size of 14.2 nm , which corresponds to feature sizes of annealed BHJ films as
determined via AFM phase images 4. All BHJ morphologies used in this study had a feature
size of 14.2 nm. Simulated structures with a feature size of 14.2 nm did not have any isolated
islands which is in accordance with expectation since the donor and acceptor phase volumes
of a blend with a 1:1 mixing ratio exceed the percolation threshold of 31 % 5. The
P3HT:PCBM composition at the top or bottom contact of the generated structures is random.
As such, statistically any interface composition could be generated if enough morphologies
are created.
2
FIG. S1. BHJ morphologies at various stages of the swapping algorithm. The feature size
increases from left to right.
S-B. Cut-off approximation for Coulomb interaction
The cut-off approximation for Coulomb interaction has been validated for. Since the electric
field in devices exhibiting s-shaped I-V curves it was necessary to determine how the I-V
curve differs when changing the cut-off radius. Fig. S2 shows J-V curves of a virtual device
that exhibits s-shaped I-V behaviour due to electron traps at the electrode interfaces. When
the cut-off radius is chosen to be too small (e.g. 10 nm) the shape of the I-V curve changes
significantly. However, for radii ≥ 15 nm no statistical difference is seen between I-V curves.
Hence, a cut-off radius equal to the Coulomb capture radius, which in this case is 19 nm, does
not affect the shape of the I-V curve.
3
FIG. S2. J-V curves of a simulated device with electron traps at the electrode to induce sshape behaviour. The cut-off radius was varied as per the legend.
S-C. Effect of energy traps on exciton transport
Energy traps in the energy landscape will not only affect charge carriers, but also
excitons. Energetic disorder is known to affect the exciton diffusion length and thus the
exciton dissociation efficiency6,7. It is important to quantify the impact of energy traps on
exciton transport in order to delineate excitonic effects from charge related issues. Exciton
dissociation at the donor-acceptor interface is very efficient (near unity). Consequently, one
may define the exciton dissociation efficiency as the number of excitons that reach the donoracceptor interface divided by the total number of excitons created in the same amount of
time. If the trap density is high, many traps will be situated directly at the heterointerface.
Dissociation at this interface involving a trap site may not occur with unity efficiency,
because the energy levels at such a site is likely not to be conducive towards exciton
dissociation. Therefore, in the following we introduce a new term, ηexint, to distinctly
differentiate between actual exciton dissociation and the probability of exciton to reach the
heterointerface. ηexint is defined as the number of excitons that reach the donor-acceptor
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interface divided by the total number of excitons created in the same amount of time. ηexint is
synonymous with exciton dissociation efficiency when the trap density is low. Simulations
indicate that the diffusion length and thus ηexint decreases with increasing trap density and
depth, see Fig. S3(a) and (b).
FIG. S3. (a) ηexint as a function of trap depth for a trap density of 0 (blue), 6.25 x 1023 (red),
1.25 x 1024 (black) and 2.50 x 1024 m-3 (green). The dashed and solid lines correspond to R0 =
3.65 nm (1D diffusion length = 40 nm) and R0 = 1.8 nm (1D diffusion length = 5 nm) for
excitons in the electron acceptor material respectively. In both sets of simulations, R0 = 1.8
nm (1D diffusion length = 8.5 nm) for excitons in the electron donor material. (b) ηexint as a
function of trap density for traps with an energy depth of 1 eV. The blue line corresponds to
R0 = 1.8 nm (i.e. PCBM8) and the red line corresponds to R0 = 3.65 nm (i.e. C609).
Fig. S3(a) shows ηexint as a function of dtrap. As traps become deeper than ~0.4 eV, the
diffusion length does not decrease anymore and ηexint levels off. This plateau exists because
the escape rate for traps deeper than ~0.4 eV is smaller than 1/lifetime. Not only the lifetime
dictates at what point the curve levels off, but also the exciton diffusion length, because the
5
latter influences the escape rate for excitons out of traps. In these simulations exciton hopping
parameters associated with P3HT (8.5 nm 10) and PCBM (5 nm8) were used. Another set of
simulations were run where the diffusion length of acceptor moiety was changed to 40 nm,
corresponding to C60 9. Compared to PCBM ηexint increased for all values of dtrap and a trend
similar to that shown in Fig. S3 was seen. In accordance with expectations, the point at which
the curve levels off is shifted towards deeper traps for larger diffusion lengths. For the range
of trap densities and depths presented in Fig. S3 the relative drop in ηexint does not exceed 5
%. Since exciton transport is assumed not to be influenced by the electric field, exciton trap
effects cannot produce s-shaped I-V curves. However, an overall drop of 5 % in current
across the voltage range may be expected when a significant amount traps are located in the
electron acceptor moiety.
S-D. Fixed trap depth simplification
Not considering a distribution of energy levels for the energy trap population removes
two extra variables (type of distribution and standard deviation of the distribution). This
approach is adequate for a general consideration of the effect of energy traps on the shape of
I-V curves as modelled for the first time with an OPV DMC model if the trap density is low
enough to be considered a perturbation to the Gaussian disorder. For trap densities as
reported in literature (1022 - 1024 m-3
11,12
), the trap population can be considered a small
perturbation as illustrated by the histograms in Figure S4. For a trap density of 1022 m-3, the
trap population is not even visible and the Gaussian shape of the histogram is entirely
preserved. For the highest trap density only a small bump is seen at -0.2 eV and the Gaussian
character of the energy landscape remains preserved.
6
FIG. S4. Histograms of the energy landscape due to energetic disorder and traps with a
density of (a) 1022 m-3 and (b) 1024 m-3. The trap depth was 0.2 eV.
Since traps have such a small impact on the total energetic character, not assigning a
distribution to the traps is a reasonable approximation. The validity of this approximation is
further demonstrated by the fact that s-shaped I-V curves were successfully simulated (see
Section A of Results and Discussion). However, the distribution simplification may give rise
to unrealistic effects at high trap densities, because the addition of traps with fixed depth
essentially removes energetic disorder. At high trap densities, the standard deviation of the
energetic landscape, σ, is indeed significantly affected. In this case, the trap population can
no longer be considered a small perturbation and within the model excitons can even hop
between traps. In order to see what trap densities are needed to see these effects, the same
exciton simulations of Fig. S3 were run with trap densities of 6.25 x 1025, 1.25 x 1026 and
2.50 x 1026 m-3. To put these values into perspective, at these densities 12.5 %, 25 % and 50
% of the available electron acceptor sites are occupied by energy traps respectively as
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opposed to 0 %, 0.125 %, 0.25 % and 0.5 % for the data presented in Fig. S3. Fig S5 shows
the results corresponding to the high trap density simulations. Here, the counteracting effects
of slowing down of excitons due to traps and the effect of an increase in exciton mobility due
to a decrease in energetic disorder are seen. These two effects almost nullify each other for
C60 (dashed lines) as there is not much variation with dtrap or ntrap. For the results
corresponding to PCBM and the highest trap density, it is seen that ηexint is even consistently
larger compared to having no traps irrespective of trap depth. The results corresponding to
ntrap < 6.25 x 1025 m-3 do not deviate much from the trend as seen for the low trap densities.
In other words, for realistic trap densities as are reported in literature (1022 – 1024 m-3 11,12) no
anomalous behaviour is seen and thus the simplified approach is justified.
FIG. S5. (a) ηexint as a function of trap depth for a trap density of 0 (blue), 6.25 x 1025 (red),
1.25 x 1026 (black) and 2.50 x 1026 m-3 (green). The dashed and solid lines correspond to R0 =
3.65 nm (C60) and R0 = 1.8 nm (PCBM) for excitons in the electron acceptor phase
respectively. In both sets of simulations, R0 = 2.15 nm (P3HT) for excitons in the electron
8
donor phase. (b) ηexint as a function of trap density for traps with an energy depth of 1 eV. The
blue line corresponds to P3HT:PCBM and the red line corresponds to P3HT:C60.
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