Exciton and Charge Dynamics in Polymer Solar Cells Studied by Transient Absorption Spectroscopy - Organic Solar Cells: Materials and Device Physics - page 129

T / K

3 00 250 200 150

(a) -25

n

-30

-35

-25

(b)

n

-30

-35

3 4 5 6 7

1000 T −1 / K −1

Fig. 5.21 Arrhenius plots of the bimolecular recombination rate c at a 1000 nm and b 700 nm
over the temperature range from 150 to 290 K. The carrier density n is 1.3 9 10 16 (r),
2.7 9 10 16 (e), 5.4 9 10 16 (j), 1.1 9 10 17 (h), 1.6 9 10 17 (d), and 2.2 9 10 17 cm -3 (s)at
700 nm and 1.3 9 10 16 (r), 2.5 9 10 16 (e), 5.0 9 10 16 (j), 1.0 9 10 17 (h), 1.5 9 10 17 (d),
and 2.0 9 10 17 cm -3 (s) at 1000 nm from bottom to top in each panel. The solid lines represent
fitting curves with the Arrhenius equation: ln c = ln A-E A /(k B T). Adapted with permission from
[ 20 ]. Copyright 2010 American Chemical Society

dn ð t Þ
dt c 0 n 3 ð t Þ

ð 5 : 5 Þ

This is consistent with the trimolecular recombination dynamics reported recently
for RR-P3HT:PCBM blends [ 67 , 68 , 71 - 73 ]. More specifically, c(n) can be
expressed by c(n) n (1/a)-1 where the exponent (1/a) - 1 would vary from
4.3 to 2.4, because a varies from 0.3 to 0.7 depending on PCBM fractions or
annealing conditions as reported previously [ 8 , 16 , 67 , 68 ]. In other words, the
time-dependent trap-limited bimolecular recombination rate is strongly dependent
on the film morphology.
For delocalized polarons, on the other hand, the time-independent trap-free
bimolecular recombination rate is c & 10 -12 cm 3 s -1 , which is one order of
magnitude higher than that estimated by photo-CELIV [ 74 ]. This is probably
because the photo-CELIV measurement cannot distinguish these two polarons, and
hence gives the averaged bimolecular recombination rate. Interestingly, the trap-
free bimolecular recombination rate is several orders of magnitude lower than the
Langevin recombination rate given by c L = e(l e ? l h )/ee 0 where l e and l h are the
electron and hole mobility, respectively, and e and e 0 are the relative permittivity
of the film and the vacuum permittivity, respectively. Assuming

l e & 10 -2 cm 2 V -1 s -1 [ 75 ] and l h & 10 -4 cm 2 V -1 s -1 [ 76 ], c L is estimated

as *10 -8 cm 3 s -1 . to be as high Considering the slowest mobility

l h & 10 -4 cm 2 V -1 s -1 [ 77 ], c L is estimated to be *10 -10 cm 3 s -1 , which is