Charge transport perpendicular to the high mobility plane in organic crystals: Bandlike temperature dependence maintained despite hundredfold anisotropy

Charge carrier mobility in van der Waals bonded organic crystals is strongly dependent on the transfer integral between neighboring molecules, and therefore the anisotropy of charge transport is determined by the molecular arrangement within the crystal lattice. Here we report on temperature dependent transport measurements along all three principal crystal directions of the same rubrene single crystals of high purity. Hole mobilities are obtained from the carrier transit time measured with high-frequency admittance spectroscopy perpendicular to the molecular layers $\left({\mu }_c\right)$ and from the transfer characteristics of two field-effect transistor (FET) structures oriented perpendicularly to each other in the layers $({\mu }_a$ and ${\mu }_b)$. While the measurements of the field-effect channels confirm the previously reported high mobility and anisotropy within the $ab$ plane, we find the mobility perpendicular to the molecular layers in the same crystals to be lower by about two orders of magnitude $({\mu }_c\sim 0.2{\mathrm{cm}}^2/\mathrm{Vs}$ at $300\mathrm{K})$. Although the bandwidth is vanishingly small along the $c$ direction and the transport cannot be coherent, we find ${\mu }_c$ to increase upon cooling. We show that the delocalization within the high mobility $ab$ plane prevents the formation of small polarons and leads to the observed “bandlike” temperature dependence also in the direction perpendicular to the molecular layers, despite the incoherent transport mechanism.

Figure 1

Optical photograph of sample 1 consisting of two perpendicular FET channels with Cytop as gate dielectric and two laminated electrodes on top/bottom of the crystal forming a vertical channel. The ribbon-shaped rubrene crystal was laminated such that the $a$ and $b$ axis are aligned with the source-drain directions of the FETs. The stacking order of the rubrene molecules leads to the highest orbital overlap in the $b$ direction, for which the largest mobility was measured. The same device geometry was used for samples 2 and 3.

Figure 2

Hole mobility in rubrene in all three principle crystal directions as a function of temperature. All the mobilities are measured simultaneously and increase towards lower temperatures, although the charge transport is highly anisotropic, i.e., ${\mu }_c$ is $\sim {10}^2$ times smaller than ${\mu }_b$. The formation of cracks leads to an apparent discontinuous drop of $\mu$ and for sample 2, the ${\mu }_a$ curves are shifted correspondingly. For comparison, the gray dashed line shows the modeled temperature dependence of ${\mu }_c$, assuming a charge transfer integral $t_c=0.9\mathrm{meV}$ and including the wave function delocalization within the $ab$ planes.

Figure 3

Density of states (DOS) in the highest occupied band computed for the dynamically disordered $ab$ plane of rubrene at 300 and $200\mathrm{K}$.

Figure 4

Visualization of the charge hopping between two adjacent rubrene molecule layers. The hopping rate is given by the Fermi golden rule. Within the layers the wave function is spread out over many molecules, and charge transfer takes place on a much faster timescale than from layer to layer.

Figure 5

FET characteristics of crystal 1 to determine ${\mu }_a$ and ${\mu }_b$: (a) The output curves, here shown for the $b$ direction at $300\mathrm{K}$, have a linear zero crossing, i.e., the drain current is not contact limited. (b) The transfer characteristics exhibit a steep subthreshold slope $(S\sim 0.1\mathrm{V}/\mathrm{dec})$ and no hysteresis, indicating a low density of trap states. (c),(d) In both the $b$ and $a$ direction the slope of the square root of $I_d$ increases upon cooling due to higher mobilities at lower temperatures.

Figure 6

Modeled theoretical small signal conductance of an ideal semiconductor structure sandwiched between two ohmic contacts. The analytical model (blue curve) corresponds to Eq. (6) and neglects diffusion currents. For comparison, we added a numerical solution for finite temperatures, including diffusion (yellow to red curves). All curves exhibit a first negative peak at $f\tau \sim 1.16$, yielding a relation between the transit time $\tau$ and the frequency, at which the oscillation occurs.

Figure 7

Mobility measurement in the $c$ direction of crystal 1 using admittance spectroscopy: (a) Measured conductance spectrum at $300\mathrm{K}$ for several bias voltages $V_0$, using a fixed AC voltage $V_{\mathrm{ac}}=0.2\mathrm{V}$, yielding textbooklike transit oscillations with first negative peaks $f_t$ between 6 and $40\mathrm{MHz}$. (b) For comparison: Modeled conductance spectrum of an ideal trap-free semiconductor at $300\mathrm{K}$ with a mobility of $0.1{\mathrm{cm}}^2/\mathrm{Vs}$. We obtain these results by numerically solving the drift-diffusion equations [Eq. (5) $+$ diffusion term], using only mobility, temperature, and geometry as input parameters, without any data fitting.

Figure 8

Bias dependence of the transit frequencies $f_t$ measured on crystal 1 at different temperatures. The mobility ${\mu }_c$ is calculated from the slope of $f_t\left(V_0\right)$ and increases with decreasing temperature. The linearity of $f_t\left(V_0\right)$ is a sign for field-independent mobility. The transit frequencies obtained from the simulations for $\mu =0.1{\mathrm{cm}}^2/\mathrm{Vs}$ (see Fig. 7) are shown for comparison.