Supporting information:
The frequency dependent mobility of charge carriers on a polymer chain with
finite length.
A general result for the frequency dependent mobility of charge carriers in the presence
of a small electric field has been derived by Kubo and is given by [1-4]
e 2
ac ( )  
2dkB T



x 2 (t) cost  dt
(A1)
0
with e the elementary charge,  the radial frequency of the electric field, d the
dimensionality of the system, kB Boltzmann’s constant, T the temperature and x 2 (t)
the mean squared displacement of the charge in the absence of an external electric field.
An implicit convergence factor exp t  (lim   0 ) is understood inthe integral [2, 3].
In the absence of electron-phonon interactions charge transport along a polymer chain


would be coherent. However,
decoherence occurs due to interactions between the
charge carrier and phonons, leading to diffusive transport [5-7]. This decoherence
occurs on a timescale corresponding to phonon frequencies; i.e. of the order of tens of
femtoseconds. Since this timescale is much shorter than the oscillation period (30 ps) of
the probing microwave field, the motion of charge carriers can be considered as
diffusive. Note, that describing charge transport as diffusive motion does not imply
hopping between localized states. Interactions of a delocalized charge with structural
fluctuations can lead to a linear increase of the mean squared displacement with time.
Hence, the transport of a delocalized charge can be diffusive. In the case of normal
Gaussian diffusion the mean squared displacement of charge carriers on a polymer
chain with infinite length is given by
x 2 (t)  2Dt .

(A2)
Insertion of Eq. A2 into Eq. A1 shows that the diffusion constant D is related to the
frequency independent mobility according to the familiar Einstein relation


e
D.
kB T
(A3)
In the present work the diffusion and mobility of charge carriers on polymer chains with
a finite length a is considered. Hence, the mean squared displacement and the mobility
of charge carriers are not given by Eqs. A2 and A3. However, Eq. A1 is still valid and
can be used to calculate the frequency dependent mobility from the mean squared
displacement of charge carriers on a finite polymer chain.
In close analogy with the treatment of Pearson[8], we obtain the mean squared
displacement on a linear chain by solving the one-dimensional diffusion equation
C(x,t)
 2C(x,t)
D
.
t
x 2
(A4)
In Eq. A4 the charge density is denoted by C, and depends on position (x) and time (t).
The ends of the polymer chain act as reflecting boundaries for the charge density. This
leads to the following conditions
 C(x,t)
0
 x x0
 C(x,t)
0
 x xa
(A5)
where x=0 and x=a denote the polymer chain ends. For a charge initially (t=0) located
at x=q the charge density is given by
1 
 kx
 kq ( ak )2 Dt
.
C(x,t)   cos
cos
e
a k  
a
a
(A6)
The average time dependence of the mean squared displacement follows by integrating
the squared displacement over all positions with their respective weight, determined by
Eq. A6; i.e.
a
(x  q)  
2
 dx(x  q) C(x,t) 
2
0

a
 kq  
q
q  ( ak )2 Dt
 a  
k

 aq  q 2  4     cos
(1

)(1)

e

 k 
3
a 
a
a
k 1
2
2
(A7)
To obtain the mean squared displacement of the charges as a function of time we
assume that the initial position of the charge carriers is uniformly distributed along the
polymer chain, thus we average Eq. A7 over all initial sites, i.e. q [0,a]
a
x (t)  (x  q)   a
2
2
1

0
4
c

 1   2  ( ak )2 Dt 

dq(x  q)   a      e

6 k  0  ck 




2
2
(A8)
where ck  2 (k 1 2)[9]. The frequency dependent mobility  ac ( ) results upon
substitution of Eq. A8 in Eq. A1

ac ( ) 
8eD 

kBT k  0
ck2
 c2 D 
1   k 2
 a 
2
.
(A9)
Eq. A9 describes the frequency dependent mobility of charge carriers that diffuse along
a polymer chain with finite length. The diffusion constant of the charge carriers is
related to the intra-chain mobility by Eq. A2; i.e. intra  eD/kBT . The frequency
dependent mobility calculated with Eq. A9 is shown in Fig. A1 as a function of intra-
 (Fig. A1b) and probing frequency (Fig.
chain mobility (Fig. A1a), polymer chain length
A1c).
In the main text the dependence of the high frequency mobility on the intra-chain
mobility is discussed in terms of the diffusive motion of the charge that is limited by the
chain ends. Alternatively, the motion of the charge can be discussed by considering the
effect of an external electric field on the charge distribution. For a polymer chain with
finite length in a static external electric field the charges will be distributed according to
a Boltzmann distribution (the equilibrium distribution). The average displacement of the
charges (equilibrium displacement) induced by the electric field is determined by the
length of the polymer chain and the strength of the electric field. The timescale on
which the equilibrium is reached is determined by the equilibrium displacement and the
charge carrier mobility (  intra ). In an oscillating electric field E( ,t)  E0 cos( t) the
charges are continuously attempting to reach the (continuously varying) equilibrium
distribution. As a consequence the average displacement of the charge carriers
 x   [ in-phase E0 sin( t)  out-of-phase E0 cos( t)] / 
v
and
the
average
velocity
dx
 in-phase E0 cos( t)  out-of-phase E0 sin( t) are partly in-phase and out-ofdt
phase with the oscillating electric field. In our experiments the high-frequency mobility
is determined by the microwave energy that is absorbed due the motion of the charge
carriers, or equivalently, the amount of work performed by the external electric field on
the charge carriers. This work is given by the product of the velocity and the electric
field according to W  e  v E dt . Hence only the in-phase component of the velocity
contributes to the absorbed energy and, consequently, to the high-frequency mobility.
As the intra-chain mobility increases both the amplitude of the average displacement
of the charges, and the phase difference between the average velocity and the electric
field increase. The increase in average displacement results in an increase in the energy
that is absorbed by the charges. The increase of the phase difference of the average
velocity and the electric field leads to a reduction of the in-phase component of the
velocity and results in a decrease of the absorbed energy. The combination of these
factors leads to the dependence shown in Fig. A1a. If the intra-chain mobility is
sufficiently small so that the charge carrier distribution is far from the equilibrium
distribution, ac   is equal to  intra . This corresponds to the limit of an infinitely long
polymer chain where the charge carrier displacement is not limited by the chain length.
In the limit of an infinitely high  intra the charges instantaneously reach the equilibrium
distribution. In this case the position of the charges is completely in phase with the field,
while the velocity is 90 degrees out of phase with the field, leading to no energy
absorption and zero value of ac  .
Fig. A1b shows ac   as a function of the polymer chain length. For short chains
the equilibrium displacement of charges corresponding to the equilibrium distribution is
small. Hence, for shorter chains the average displacement of the charges is more in
phase with the probing field and the average velocity is more out of phase with the field.
As a consequence ac   is smaller for shorter chains. For longer chains the
equilibrium displacement is larger. Consequently the average displacement of the
charges lags behind the oscillating field. Hence the velocity of the charges is more in
phase with the field, leading to an increase in absorbed energy or equivalently an
increase of ac  . In the limit of an infinitely long chain the average velocity is
exactly in phase with the probing field and ac   is equal to  intra .
Fig. A1c shows the dependence of ac   on the frequency of the probing electric
field. The phase-lag of the average displacement with respect to the probing field
increases with frequency. Since the average velocity of the charges is more in phase
with the probing field for higher frequencies, ac   increases with frequency. For
very high frequency the average displacement becomes 90 degrees out of phase with the
driving field and the velocity of the charges is in phase with the field, leading to
maximal energy absorption. This corresponds to the limit of an infinitely long polymer
chain where the displacement of the charge carriers is not limited by the chain length
and ac   is equal to  intra . As can be seen in Eq. A9 the dependence of ac   on a2
is equivalent to the dependence on  . Comparison of Figs. A1b and A1c shows that
doubling the polymer chain length has the same consequence for the high-frequency
mobility as increasing the frequency fourfold.
It might be argued that the limited chain length of the polymer chains gives rise to a
dependence of the intra-chain mobility on the magnitude of the probing electric field. In
order to investigate whether this effect plays a role in the current experiment (where a
field strength of 20 V/cm is used) the high-frequency mobility was also obtained from
Monte Carlo computer simulations of the diffusion and drift of charge carriers on finite
chains, subject to an oscillating external electric field E(t)=E0cos(wt). The displacement
of a single charge carrier during a small time step Dti was calculated as [10]
xi  intra E t ti  x diff ,i

(A10)
The first term at the right-hand side of Eq. A10 accounts for the drift motion in the
electric field. The second term is the random diffusive displacement x diff,i , which is
2
sampled from an interval such that x diff
,i 2Dti . Reflection of the charge carriers

at the polymer chain ends was taken into account. The high-frequency mobility was
 of work performed by the external electric field on the
calculated from the amount
charge carriers [11]. The work performed during a series of time steps Dti in the
simulations can be calculated as W   eE ti xi . On the other hand, the high–
i


frequency mobility is given by ac  W  e E 2 t  dt . Taking the time interval
0
 
 n2 / , with n an integer, yields ac  2W / eE 02 . Using this expression, the

high-frequency mobility can be obtained from simulations of the motion of the charge

carriers and monitoring the 
work done by the field during time. The high-frequency
mobility obtained from the simulations was found to agree with that calculated from Eq.
A9 for external electric field strengths up to 2x103 V/cm, which is two orders of
magnitude higher than the field strength used in the experiments. Hence, in the
experiments non-linear effects due to the presence of polymer chain ends do not play a
role.
Supporting information:
Truncated Flory distribution
The chain length distribution of each of the four the ladder-type polymers is known to
be a Flory distribution, where the probability of a chain consisting of n monomers is
given by P(n) = (1-p)pn-1, where p=(<n>-1)/<n> denotes the probability of bond
formation between adjacent monomers during the synthesis of the polymer. Chains
shorter than five monomers have been removed by solvent extraction and the presence
of chains longer than 75 monomers is unlikely due to the method of synthesis.
References
[1]
R. Kubo, J. Phys. Soc. Jap. 12, 570 (1957).
[2]
H. Scher and M. Lax, Phys. Rev. B 7, 4491 (1973).
[3]
J. C. Dyre and J. M. Jacobsen, Chem. Phys. 212, 61 (1996).
[4]
U. Mizutani, Introduction to the Electron Theory of Metals (Cambridge
University Press, Cambridge, 2001).
[5]
P. Reineker, Z. Physik 261, 187 (1973).
[6]
M. A. Palenberg, R. J. Silbey, and W. Pfluegl, Phys. Rev. B 62, 3744 (2000).
[7]
E. A. Silinsh, Organic Molecular Crystals (Springer-Verlag, Berlin, 1980).
[8]
D. S. Pearson, P. A. Pincus, G. W. Heffner, and S. J. Dahman, Macromolecules
26, 1570 (1993).
[9]
(note, that Eq. 5 in reference 8 is flawed).
[10]
H. Risken, The Fokker-Planck Equation (Springer-Verlag, Berlin, 1984).
[11]
O. Hilt and L. D. A. Siebbeles, Chem. Phys. Lett. 269, 257 (1997).
Fig. A1. High-frequency mobility calculated with Eq. A9 as a function of a) intra-chain
mobility (f = 30 GHz, n = 100), b) number of repeat units in the polymer chain (f = 30
GHz,  intra = 100 cm2/Vs) and c) frequency (n = 100,  intra = 100 cm2/Vs).