Microscopic origins of charge transport in triphenylene systems

We study the effects of molecular ordering on charge transport at the mesoscale level in a layer of $\approx 9000$ hexa-octyl-thio-triphenylene discotic mesogens with dimensions of $\approx 20\times 20\times 60{\mathrm{nm}}^3$. Ordered (columnar) and disordered isotropic morphologies are obtained from a combination of atomistic and coarse-grained molecular-dynamics simulations. Electronic structure codes are used to find charge hopping rates at the microscopic level. Energetic disorder is included through the Thole model. Kinetic Monte Carlo simulations then predict charge mobilities. We reproduce the large increase in mobility in going from an isotropic to a columnar morphology. To understand how these mobilities depend on the morphology and hopping rates, we employ graph theory to analyze charge trajectories by representing the film as a charge-transport network. This approach allows us to identify spatial correlations of molecule pairs with high transfer rates. These pairs must be linked to ensure good transport characteristics or may otherwise act as traps. Our analysis is straightforward to implement and will be a useful tool in linking materials to device performance, for example, to investigate the influence of local inhomogeneities in the current density. Our mobility-field curves show an increasing mobility with field, as would be expected for an organic semiconductor.

Figure 1

Top panels: top (left) and side (right) views of the 2,3,6,7,10,11-hexahexyl thiotriphenylene (HHTT) molecule ($R=\text{octyl}$) corresponding to the central core of HOTT with the chains replaced by hydrogen. The oblate Gay-Berne ellipsoid for coarse-grained simulations is also shown. The radius of an HHTT molecule in the $x-y$ plane, $r_{\perp }\approx$ 0.653 nm, and radial height, $r_{\parallel }\approx$ 0.172 nm. Bottom panels: views along (left) and across (right) the column axis direction from atomistic packing simulations.

Figure 2

Schematic illustration of the network. Nodes are the molecules shown as ellipses. For each pair of molecules separated by less than $r_c$ there are two edges whose directions are shown by the arrows joining the nodes. The node labels show how the linked traffic ${\widehat{T}}_{ij}$ is calculated from Eq. (13).

Figure 3

HOTT morphologies after back-mapping to atomic coordinates. (a) Disordered morphology generated at $T=280\mathrm{K}$, system size is $15.67\times 15.67\times 52.85\mathrm{nm}$. (b) Columnar morphology generated at $T=400\mathrm{K}$, system size is $16.32\times 16.32\times 65.36$ nm.

Figure 4

Pair correlation function $g$ vs molecular separation $r$ (nm) or HOTT molecules in columnar and disordered phases. For the former, the first peak is at 0.43 nm and represents the intracolumn vertical separation, the second peak is at 0.904 nm, while the first minimum is at 0.74 nm. The first peak in the disordered phase is at 0.515 nm.

Figure 5

The distribution of hole transfer integrals $J_{ij}^2$ between pairs of molecules with respect to the Cartesian components of the pair separation, according to $r_{ij}^2=\mathrm{\Delta }{x}_{ij}^2+\mathrm{\Delta }{y}_{ij}^2+\mathrm{\Delta }{z}_{ij}^2$ (nm). (a) Columnar phase. The dashed red line illustrates the variation of $J^2$ when moving along a column. (b) Disordered phase.

Figure 6

Charge mean-squared displacement $〈r_z^2〉$ (solid lines) and $〈z_z^2〉$ (dotted lines) vs simulation time $\tau$ (ps) for columnar and disordered phases. Inset: a subsample of the same data on a linear scale.

Figure 7

Simulation cells where $0\le x\le x_{\text{max}}, 0\le y\le y_{\text{max}}, 0\le z\le z_{\text{max}}$, and $x_{\text{max}},y_{\text{max}},z_{\text{max}}$ are given in the caption to Fig. 3. (a) and (b) Edges with transfer integrals $|J_{ij}{|}^2>{10}^{-6}\mathrm{eV}$ shown as arrows; (c) and (d) traffic $\mathbf{T}$ between pairs (lines); (e) and (f) the extensive flux $\mathbf{F}{\tau }_{\max }$ integrated over ${\tau }_{\max }$ in columnar and disordered phases. The grids are to guide the reader's eye.

Figure 8

Cluster size vs cluster length along the $z$ axis (the direction of an applied bias of $2\mathrm{V}$) for the columnar phase at simulation times of ${10}^5$ (blue diamonds) and ${10}^7$ ps (circles) and for the disordered phase at ${10}^7$ ps (red squares).

Figure 9

Distribution of $J_{ij}$ in the columnar and disordered phases. For the former there is a clear bimodal distribution due to the high spatial and orientational ordering of the molecules. For the latter, the distribution is flatter. The solid line is the distribution taken over all molecule pairs up to 2.5 nm apart; the dashed line is for pairs up to 1 nm. Inset: the distribution of pair separations in the two phases.

Figure 10

The traffic between two molecular sites vs the linked traffic of that pair as defined in Eq. (13). Both measures of the traffic are more tightly distributed in the ordered system.

Figure 11

Poole-Frenkel mobility-field plot for columnar and disordered phases. The mobility increases with increasing applied field in both systems, although the response is much stronger in the columnar system. The weak dependence upon the applied field in the amorphous system is consistent with Figs. 5 and 6; the field is more effective when aligned to the coupling topology.