Page 41 - Organic Electronics in Sensors and Biotechnology

18 Chapter One

In a disordered organic semiconductor system, charge transport
occurs mainly by hopping between adjacent or nearby localized states
which are induced by disorder. The physical effect of the longitudinal
electrical field is then to effectively reduce the hopping barrier. From
this basic concept and the reasonable assumption of Coulomb poten-
tial type for hopping barrier, the hopping probability, and therefore
the mobility, will demonstrate a dependence on electrical field which
follows a Frenkel-Poole relationship 64, 65

⎛ ⎞
μ = μ exp ⎜ β E ⎟ (1.1)
0
⎝ kT ⎠
where k = Boltzmann’s constant
T = temperature
E = electrical field
μ = mobility at zero field
0
β= field-dependent coefficient originally proposed by Gill in
an empirical relation 66
This law predicts the experimental results represented by the four
straight dashed lines in the panels of Fig. 1.10. Field dependence of
mobility becomes more severe at lower temperature, as Fig. 1.10 indi-
cates and Frenkel-Poole’s law predicts. When temperature decreases,
the field dependence of the mobility becomes stronger as indicated in
Eq. (1.1); namely, the slope of the logarithmic mobility-field curve in
Fig. 1.10 is steeper at lower temperature for the same channel length
under the same longitudinal field. The temperature-dependent beha-
vior of the field-dependent mobility for charge transport in these
organic field-effect transistors is well exhibited in Fig. 1.11, from 44 K
to room temperature. Figure 1.11 is a collection of fitting lines each of
which represents the behavior of field-dependent mobility at a certain
temperature, obtained from experimental data in the same way as the
four straight dashed lines in Fig. 1.10. It is interesting and important to
notice the converging point in Fig. 1.11; i.e., at the field ~7.3 × 10 V/cm,
5
2
mobilities at all temperatures fall onto the same value ~0.15 cm /(V ⋅ s),
corresponding to a zero hopping barrier at such a high field. This is
also predicted by Frenkel-Poole’s model. Recalling that zero-field
mobility μ in Eq. (1.1) can be expressed as
0
⎛ Δ ⎞
μ = μ exp − (1.2)
0 i ⎜ ⎝ kT ⎟ ⎠
65
based on the basic description for hopping transport, the Frenkel-
Poole’s expression for field-dependent mobility can be comprehen-
sively written as

⎛ − Δ ⎞
μ = μ exp ⎜ β E
i ⎟ (1.3)
⎝ kT ⎠