Charge transport in disordered semiconducting polymers driven by nuclear tunneling

The current density-voltage ($J-V$) characteristics of hole-only diodes based on poly(2-methoxy, 5-(2′ ethyl-hexyloxy)-$p$-phenylene vinylene) (MEH-PPV) were measured at a wide temperature and field range. At high electric fields the temperature dependence of the transport vanishes, and all $J-V$ sweeps converge to a power law. Nuclear tunneling theory predicts a power law at high fields that scales with the Kondo parameter. To model the $J-V$ characteristics we have performed master-equation calculations to determine the dependence of charge carrier mobility on electric field, charge carrier density, temperature, and Kondo parameter, using nuclear tunneling transfer rates. We demonstrate that nuclear tunneling, unlike other semiclassical models, provides a consistent description of the charge transport for a large bias, temperature, and carrier density range.

Figure 1

The charge carrier mobility dependence for three different hopping models. Miller-Abrahams hopping and Marcus hopping both predict decreasing values for increasing fields, while nuclear tunneling hopping (${\alpha }_K=2$) shows an increasing mobility. The open symbols correspond to a charge density of ${10}^{-4}{a}^{-3}{\mathrm{m}}^{-3}$, while the solid symbols represent data from ${10}^{-2}{a}^{-3}{\mathrm{m}}^{-3}$. On the $x$ axis, the electric field is divided by the product of the lattice constant and the elementary charge. The inset represents the electron transfer in a biased double quantum well. ${\epsilon }_{ij}$ is the difference between the minima of the potential energy wells, $\lambda$ is the reorganization energy, and $H_{ij}$ is the tunnel splitting. The solid red arrow indicates the semiclassical Marcus hopping path while the dashed red arrow represents nuclear tunneling.

Figure 2

Dependence of mobility on temperature and carrier density for ${\alpha }_K=2$ and $F=0.01\sigma a^{-1}{q}^{-1}\mathrm{V}{\mathrm{m}}^{-1}$. The lines are given by Eq. (4). ${\mu }_0$ is given by $\frac{{\nu }_0{a}^{2}q}{\sigma }$. The inset shows the electric field dependence of the mobility for two values of ${\alpha }_K$: For large fields, the density dependence vanishes and the mobility is given by Eq. (5).

Figure 3

Current-voltage characteristics of MEH-PPV hole-only diodes with thicknesses of 40, 80, and 115 nm measured at temperatures ranging from 175 to 300 K. The lines correspond to drift-diffusion simulations based on a nuclear tunneling mobility model. All fits use the same values for ${\alpha }_K, \sigma , a$, and ${\nu }_0$.