Fig. 5: Influence of spatial parameters and intrinsic disorder on the calculated activation energy of conductivity for F₈ZnPc.

a This plot underlines that the spatial and energetic aspects of transport are closely connected: Halving the localization radius α (blue circles) compared to a reference sample (green squares), i.e. doubling the penalty of long-distance hops, also increases the activation energy by up to 100 meV. This is not obvious, since the exponent of the hopping rate (Eq. (6)) that contains α does not even depend on temperature. The connection of α and E_A only emerges due to the tradeoff of variable-range hopping: For stronger localization, low-energy hops are more likely to be traded for short-distance hops. Since the typical critical hopping distance is the distance between dopants at low doping ratios, lowering the dopant distance is the main reason for the decrease in activation energy upon increasing doping ratio. Interestingly, decreasing α is not equivalent to increasing the lattice distances d (black triangles) at low and moderate doping ratios. At high concentrations, however, where the charges get very close and the typical hopping distance saturates at the lattice distance, the effect of halving α and of doubling d becomes equivalent. b E_A versus molar doping ratio for F₈ZnPc with different widths σ₀ of the intrinsic energetic disorder. Remarkably, the sample with the highest intrinsic disorder exhibits the lowest E_A at low doping concentrations. The reason are matrix sites that are randomly lowered in energy by the intrinsic disorder and can serve as bridging sites between the dopant wells (see Fig. 3). This benefit disappears at high doping concentrations, where the notion of individual dopant wells vanishes. Error bars show one standard deviation.