technique currently available that can predict the regime of charge transport for a
given material system. To explore the limitations associated with simulating charge
transfer in a frozen morphology, Poelking et al. [13] have compared charge escape
times τ esc with relaxation times of the backbone, as reflected both in the electronic
coupling elements and in site energies. The escape time (i.e., the average time a
charge spends localized on a given site) is the inverse of the escape rate,
τ
i
ð Þ
esc
¼ 1=Γ
i
ð Þ
esc
, where Γ
i
ð Þ
esc
¼
X
j i
ð Þ
Γ ij , Γ ij is the hole-transfer rate from site i to site
j, and the sum is evaluated for all nearest neighbors j of site i. From the resulting
distribution of escape times, p(t), one can calculate the distribution (exceedence, or
complementary cumulative distribution function), P(τ) ¼
Ð 1
τ p(t)dt, which is proportional to the number of sites with an escape time larger than τ.
Backbone dynamics can be estimated from the time autocorrelation functions
R U (τ) for site energies U
(i) and R J (τ) for couplings |J ij |
2 :
R U τ
ð Þ ¼
U
i
ð Þ
t À U
h i
U
i
ð Þ
tþτ À U
h i
D
E
σ 2
,
ð40Þ
where the outer h . . . i denotes the ensemble average. The width σ and average hUi
have the same meaning as in the electronic density of states (see Sect. 4.4). An
analogous expression is used for the transfer integrals.
The autocorrelation function and the distribution of escape times for P3HT are
summarized in Fig. 11. Relaxation of the electronic coupling elements and of the
site energies occur on similar time scales in spite of their dissimilar physical
origins: Site energies are related to long-range electrostatic interactions where
averaging occurs over a large number of nearest neighbors and leads to spatial
correlations. On the other hand, the electronic coupling elements (to a first approximation) only depend on the geometries of pairs of molecules, which results in
increased sensitivity to thermal motion of the internal degrees of freedom. The
reason for similar time scales is the chemical structure of P3HT: Every thiophene
unit is linked to an alkyl side chain with slow dynamics both in the crystalline and
amorphous phases. This overdamps the backbone dynamics, particularly torsional
motions of thiophene units, and results in slow variations of electronic couplings.
Interestingly, for the similar conjugated polymer PBTTT, where the
thienothiophene unit is not linked to a side chain (implying a lower side-chain
density and better crystallinity), the significantly faster dynamics of electronic
couplings can boost the charge-carrier mobility [119].
Comparing the 100% and 90% regioregular materials, we can see how the
defects in side-chain attachment lead to slower dynamics, with decorrelation
times significantly increased over the defect-free case. More quantitatively, for
intermediate delay times in the range of tenths of picoseconds, the time evolution is
in all cases governed by a logarithmic diffusion-driven decorrelation of both site
energies and electronic couplings. Regarding site energies, the dimensionless
exponent that characterizes this decorrelation for CA-100 assumes a value three
times larger than for CA-90. This again highlights how defects in regioregularity
168
C. Poelking et al.
given material system. To explore the limitations associated with simulating charge
transfer in a frozen morphology, Poelking et al. [13] have compared charge escape
times τ esc with relaxation times of the backbone, as reflected both in the electronic
coupling elements and in site energies. The escape time (i.e., the average time a
charge spends localized on a given site) is the inverse of the escape rate,
τ
i
ð Þ
esc
¼ 1=Γ
i
ð Þ
esc
, where Γ
i
ð Þ
esc
¼
X
j i
ð Þ
Γ ij , Γ ij is the hole-transfer rate from site i to site
j, and the sum is evaluated for all nearest neighbors j of site i. From the resulting
distribution of escape times, p(t), one can calculate the distribution (exceedence, or
complementary cumulative distribution function), P(τ) ¼
Ð 1
τ p(t)dt, which is proportional to the number of sites with an escape time larger than τ.
Backbone dynamics can be estimated from the time autocorrelation functions
R U (τ) for site energies U
(i) and R J (τ) for couplings |J ij |
2 :
R U τ
ð Þ ¼
U
i
ð Þ
t À U
h i
U
i
ð Þ
tþτ À U
h i
D
E
σ 2
,
ð40Þ
where the outer h . . . i denotes the ensemble average. The width σ and average hUi
have the same meaning as in the electronic density of states (see Sect. 4.4). An
analogous expression is used for the transfer integrals.
The autocorrelation function and the distribution of escape times for P3HT are
summarized in Fig. 11. Relaxation of the electronic coupling elements and of the
site energies occur on similar time scales in spite of their dissimilar physical
origins: Site energies are related to long-range electrostatic interactions where
averaging occurs over a large number of nearest neighbors and leads to spatial
correlations. On the other hand, the electronic coupling elements (to a first approximation) only depend on the geometries of pairs of molecules, which results in
increased sensitivity to thermal motion of the internal degrees of freedom. The
reason for similar time scales is the chemical structure of P3HT: Every thiophene
unit is linked to an alkyl side chain with slow dynamics both in the crystalline and
amorphous phases. This overdamps the backbone dynamics, particularly torsional
motions of thiophene units, and results in slow variations of electronic couplings.
Interestingly, for the similar conjugated polymer PBTTT, where the
thienothiophene unit is not linked to a side chain (implying a lower side-chain
density and better crystallinity), the significantly faster dynamics of electronic
couplings can boost the charge-carrier mobility [119].
Comparing the 100% and 90% regioregular materials, we can see how the
defects in side-chain attachment lead to slower dynamics, with decorrelation
times significantly increased over the defect-free case. More quantitatively, for
intermediate delay times in the range of tenths of picoseconds, the time evolution is
in all cases governed by a logarithmic diffusion-driven decorrelation of both site
energies and electronic couplings. Regarding site energies, the dimensionless
exponent that characterizes this decorrelation for CA-100 assumes a value three
times larger than for CA-90. This again highlights how defects in regioregularity
168
C. Poelking et al.