Electric field dependence of charge carrier hopping transport within the random energy landscape in an organic field effect transistor

We extended our analytical effective medium theory [Phys. Rev. B 81, 045202 (2010)] to describe the temperature-dependent hopping charge carrier mobility at arbitrary electric fields in the large carrier density regime. Special emphasis was made to analyze the influence of the lateral electric field on the Meyer–Neldel (MN) phenomenon observed when studying the charge mobilities in thin-film organic field-effect transistors (OFET). Our calculations are based on the average hopping transition time approach, generalized for large carrier concentration limit finite fields, and taking into account also spatial energy correlations. The calculated electric field dependences of the hopping mobility at large carrier concentrations are in good agreement with previous computer simulations data. The shift of the MN temperature in an OFET upon applied electric field is shown to be a consequence of the spatial energy correlation in the organic semiconductor film. Our calculations show that the phenomenological Gill equation is clearly inappropriate for describing conventional charge carrier transport at low carrier concentrations. On the other hand a Gill-type behavior has been observed in a temperature range relevant for measurements of the charge carrier mobility in OFET structures. Since the present model is not limited to zero-field mobility, it allows a more accurate evaluation of important material parameters from experimental data measured at a given electric field. In particular, we showed that both the MN and Gill temperature can be used for estimating the width of the density of states distribution.

Figure 1

(a) Poole–Frenkel plots of electric field dependences of the effective charge carrier mobility $\ln ({\mu }_e/{\mu }_0)$ calculated by Eqs. (10, 11, 12, 13, 14) for an energy-correlated organic disordered system for several carrier concentrations ($n/N$ $=$ 10${}^{-3}$, 10${}^{-2.5}$, and 10${}^{-2}$) at $a/b=5$ and $\sigma /k_{B}T=5$ with accounting (see text) for the carrier concentration dependence of the jump length (solid curves) and ignoring the latter effect (dashed curves); (b) field dependences calculated for a noncorrelated disordered system are given for comparison.

Figure 2

Poole–Frenkel plots of electric field dependences of the effective charge carrier mobility $\ln ({\mu }_e/{\mu }_0)$ in an organic disordered system calculated by Eqs. (10, 11, 12, 13, 14) at different temperatures with accounting for energy correlations for two carrier concentrations: $n/N={10}^{-1.5}$ (upper branch) and 10${}^{-5}$ (lower branch) $a/b=5$.

Figure 3

(a) Temperature dependences of the charge carrier mobility $\ln \left({\mu }_e\right)\propto 1/T^2$ calculated by Eqs. (10, 11, 12, 13, 14) for an energy correlated system at low carrier concentration $n/N={10}^{-9}$ parametric in electric fields and at $a/b=5$; (b) the same data replotted in Arrhenius ($\ln \left({\mu }_e\right)\propto 1/T$) representation.

Figure 4

Effective charge carrier mobility $\ln ({\mu }_e/{\mu }_0)$ vs $\sigma /k_{B}T$ calculated by Eqs. (10, 11, 12, 13, 14) at finite electric field (${(eaF/\sigma )}^{1/2}=0.6$) for different effective carrier concentrations (bold curves) with accounting for energy correlations and $a/b=5$. Inset: $\ln ({\mu }_e/{\mu }_0)$ vs $\sigma /k_{B}T$ calculated at vanishing electric field ($F\to 0$) for different effective carrier concentrations (bold curves) with accounting for energy correlations. Fitting the lower curve by $\ln \left({\mu }_e\right)\propto -0.29{(\sigma /k_{B}T)}^2$ is given by the dashed curve (see text for details).

Figure 5

Effective charge carrier mobility $\ln ({\mu }_e/{\mu }_0)$ vs $\sigma /k_{B}T$ calculated by Eqs. (10, 11, 12, 13, 14) parametric in electric fields for two different carrier concentrations $n/N={10}^{-9}$ and 10${}^{-2}$ (dashed and bold curves, respectively) at $a/b=5$ in an energy correlated hopping system. The thin straight lines show extrapolation of the calculated dependences at $n/N={10}^{-2}$ to higher temperatures.

Figure 6

Calculated dependence of MN temperature $T_1$ (a) on electric field (solid curve 1) and the dependence of Gill temperature $T_2$ on carrier concentration (b) for an energy correlated disordered hopping system (solid curve 1). The same dependences calculated upon ignoring the carrier concentration dependence of the jump length are shown by dashed curves 2, and upon ignoring any correlation effects are shown by dashed curves 3.

Figure 7

Ratio of $E_G/E_{\mathrm{MN}}$ calculated by Eq. (19) for different electric fields and carrier concentrations.