Charge Transport in Guanine-Based Materials - Institut für ...

7368 J. Phys. Chem. B, Vol. 113, No. 20, 2009 Ortmann et al.
building blocks (e.g., stacked guanine molecules) to mimic basic
electronic properties of DNA but remain tractable for ab initio
methods. 15,16 For more complex assemblies such as quartet
structures 17 or the 3D guanine crystals of Figure 1, 18 it has been
shown that modifications of the electronic structure of the stack
are induced by the influence of electronic coupling perpendicular
to the stacking direction. Electronic coupling within such a single
layer is essential to understand the current along the ribbons in
the transistor experiment, as well as base pair coupling in DNA
that is a prerequisite for interstrand transport pathways (see
Figure 1). Therefore, the influence of the relative orientation of
guanine molecules deserves close attention in exploring such
structures as advanced materials. For that reason, we address
the important aspect of anisotropy effects in guanine-based
materials and charge-carrier mobilities along various directions
and planes in 3D guanine crystals.
The complexity of the charge-transport problem, however,
originates not only from the various possible pathways for the
charge carriers and the relative orientations of the molecules. It
also arises from the ambient temperature and its influence on
the carriers. 19 Whereas previous first-principles studies selected
single conformations (fixed geometry at T)0K), experiments
(mostly at room temperature) average over many of them.
However, the number of possible configurations at a given
temperature is huge and can be only crudely represented by a
single geometry at T ) 0 K. Therefore, in the present work, we
go beyond the T ) 0 K scenario by taking into consideration
vibrational degrees of freedom and the influence of their
interaction with the charge carriers. This interaction is important
because the electronic properties (e.g., the electronic interactions
between neighboring molecules) are strongly affected by
variations of nuclei positions. 4,20-22 Hence, neglecting them in
theoretical approaches prevents such approaches from being
comparable to experiment. The dependence of the electronic
properties on the geometry can be rationalized with the polaron
concept 23 based on electron-phonon interactions. By means of
the polaron concept, we take into account the change in the
geometry with the moving charge and how this is influenced
by temperature. The influence of the charge-vibration coupling
on band transport or ballistic transport was discussed earlier, 24,25
and the occurrence of polarons in DNA has been demonstrated. 26,27
Here, we go a step further and study the transport of polarons
through guanine crystals, which serve as a model system
providing multiple pathways for carriers within stacks and
across, as depicted in Figure 1. We start the discussion with
the anisotropy of the electronic coupling, followed by a
description of the polaron formation and the influence of the
temperature. Applying a general transport theory, we then
calculate the hole mobilities as a function of temperature and
direction. Finally, the transport channels are visualized to get
an unprecedented impression of the anisotropy of the carrier
mobility.
The electronic coupling between the highest occupied molecular
orbitals (HOMOs) of neighboring molecules are obtained
by first-principles density functional theory (DFT) as explained
in the Supporting Information. The resulting transfer integrals
ε MN between molecular sites M and N are compiled in Table 1.
The anisotropy in the electronic coupling is evident, as there is
c
one dominating entry: ε 11 ) 170.4 meV. This matrix element
belongs to a pair of molecules along the stacking (c) direction
(see Figure 1). It is 1 order of magnitude larger than the largest
contribution for any other direction. Note that this pair of
molecules is not perfectly cofacially aligned as in ordinary stacks
but rather is slightly shifted parallel to the molecular plane
TABLE 1: Transfer Integrals (meV) That Best Fit the
Crystal Band Structure a
integral value integral value integral value
a
ε 11 0.0
0
ε 12 -10.2
0
ε 14 1.2
b
ε 11 4.1
c
ε 12 48.2
a
ε 14 8.0
c
ε 11 170.4
- c
ε 12 5.5
c
ε 14 12.0
2c
ε 11 11.7
ac
ε 14 -5.7
bc
-1.2
ε 11
a For notation, see Figure 1.
similarly to adjacent bases in DNA. The displacement results
in a reduced transfer integral compared to a perfect stack. This
is clear from the nodal structure of the frontier orbitals and has
been described recently for organic semiconductors. 28,29 Aside
from the dominating transfer integral in stacking direction, the
electronic coupling across a hydrogen bridge to a neighboring
stack in either the a or b direction is smaller but nonzero, for
c c
example, ε 12 ) 48.2 meV or ε 14 ) 12.0 meV. In terms of
transport pathways, this makes a stack change during transport
less favorable but not impossible.
To describe temperature-induced effects in the electronic
coupling, we additionally take into account vibrational degrees
of freedom [with frequencies ω Q ≡ ω λ (q)] and the coupling
between the charges and the vibrations. We use the Hamiltonian
H)∑
[ ε MN + ∑
MN Q
pω Q g Q MN (b † Q + b -Q ) † ]a Ma N +
∑pω Q
Q( b † Qb Q +
2) 1 (1)
Q
which explicitly contains coupling constants g MN between
particles and phonons. For a moving particle, these dimensionless
parameters describe the strength of scattering by an
individual phonon mode. By means of this Holstein-Peierls
Hamiltonian, one can simulate the effect of a quasiparticle
(polaron), which is heavier than the bare charge carrier (hole)
because of the accompanying phonon cloud that has to be carried
through the system as well. This increased effective mass is
also reflected in the reduced transfer integrals for the polarons
eff
[m polaron ] -1 ∝ ε˜MN ) (ε MN - ∆ MN ) exp[-∑ λ
G λ (1+2N λ )]
where zero-point vibration effects and temperature effects
{through the phonon occupation number N λ ) [exp(pω λ /k B T)
- 1] -1 } give rise to an exponential damping factor. 30 The
λ
quantity G λ ) (g MM ) 2 + 1 / 2 ∑ K*M (g λ MK ) 2 measures the total
scattering strength of the mode λ. Additional polaron binding
terms such as ∆ MN have been discussed elsewhere. 26,31 Here,
we focus on the temperature dependence of the coupling
between different sites ε˜MN that determine the transfer rates and
hence the mobility of the charge carriers. For the guanine crystal,
we obtain effective coupling values of G λ ) 0.25, 0.09, and
0.34 for the three symmetrical rotations atpω λ ) 10.6, 15.2,
and 17.1 meV, respectively, according to the method described
in the Supporting Information. The strong influence of the
electron-phonon coupling is shown in Figure 2, where we have
plotted the effective hole polaron mass and an effective transfer
integral as obtained from eq 2. First, we recognize that, already
at zero Kelvin, the polaron mass is somewhat larger than the
bare hole mass because of zero-point vibrational effects. In other
words, the coupling seen by the electrons (ε MN ) is reduced for
the polarons (ε˜MN ). The inclusion of finite temperatures gives
rise to an even stronger increasing mass by an order of
(2)